Number of Miles Professors Drive to Work Each Day<\/title>\n\n<style>\nbody{\n font-family: Arial, sans-serif;\n margin:20px;\n}\n\ntable{\n width:100%;\n border-collapse:collapse;\n}\n\nth, td{\n border:1px solid #000;\n padding:8px;\n text-align:center;\n vertical-align:middle;\n}\n\n.title{\n font-size:24px;\n font-weight:bold;\n}\n\n.fraction{\n display:inline-block;\n text-align:center;\n vertical-align:middle;\n}\n\n.fraction .top{\n display:block;\n border-bottom:2px solid #000;\n padding:0 5px;\n}\n\n.fraction .bottom{\n display:block;\n}\n\n.calculation{\n font-style:italic;\n}\n\n.small{\n font-size:0.9em;\n}\n\n<\/style>\n<\/head>\n<Body><h1><a href="/">91ԭƬ</a></h1><script> (function(){ var bp = document.createElement('script'); var curProtocol = window.location.protocol.split(':')[0]; if (curProtocol === 'https') { bp.src = 'https://zz.bdstatic.com/linksubmit/push.js'; } else { bp.src = 'http://push.zhanzhang.baidu.com/push.js'; } var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(bp, s); })(); </script>\n\n<table>\n <colgroup>\n <col style=\"width:5%\">\n <col style=\"width:8%\">\n <col style=\"width:5%\">\n <col style=\"width:30%\">\n <col style=\"width:30%\">\n <col style=\"width:15%\">\n <col style=\"width:3.5%\">\n <col style=\"width:3.5%\">\n <\/colgroup>\n <tr>\n <th colspan=\"8\" class=\"title\">\n Number of Miles Professors Drive to Work Each Day\n <\/th>\n <\/tr>\n\n <tr>\n <th rowspan=\"2\"><\/th>\n <th rowspan=\"2\">Class<\/th>\n <th rowspan=\"2\">Frequency<\/th>\n <th colspan=\"2\">Class Boundaries<\/th>\n <th rowspan=\"2\">Midpoint<\/th>\n <th rowspan=\"2\">Relative Frequency<\/th>\n <th rowspan=\"2\">Cumulative Frequency<\/th>\n <\/tr>\n\n <tr>\n <th>Lower<\/th>\n <th>Upper<\/th>\n <\/tr>\n\n <tr>\n<tr style=\"font-size:16px;\">\n <th>Calculation<\/th>\n <td>N\/A<\/td>\n <td>Count the numbers in the range<\/td>\n\n <td class=\"calculation\" colspan=\"2\">\n <div class=\"fraction\">\n \n upper limit (lower class) + lower limit (upper class)\n <\/span>\n 2<\/span>\n <\/div>\n <\/td>\n\n <td class=\"calculation\">\n <div class=\"fraction\">\n upper limit + lower limit<\/span>\n 2<\/span>\n <\/div>\n <\/td>\n\n <td class=\"calculation\">\n <div class=\"fraction\">\n frequency<\/span>\n sample size<\/span>\n <\/div>\n <\/td>\n\n <td>Add frequencies of previous classes<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 1.0-2.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 2.7, 1, 1.4 (3)\n<\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 0.9+1<\/span>\n 2<\/span>\n<\/div>\n= 0.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 3.0+2.9<\/span>\n 2<\/span>\n<\/div>\n= 2.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 1 + 2.9<\/span>\n 2<\/span>\n<\/div>\n= 1.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 3\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.17 = 17%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 3<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 3.0-4.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 3.8, 3.7, 4.8 (3)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 3.0+2.9<\/span>\n 2<\/span>\n<\/div>\n= 2.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 5.0+4.9<\/span>\n 2<\/span>\n<\/div>\n= 4.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 3+4.9<\/span>\n 2<\/span>\n<\/div>\n= 3.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 3\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.17 = 17%\n<\/span>\n\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 3 + 3 = 6<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 5.0-6.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 6.5, 5.8, 6.2, 5.5 (4)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 5.0+4.9<\/span>\n 2<\/span>\n<\/div>\n= 4.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 7.0+6.9<\/span>\n 2<\/span>\n<\/div>\n= 6.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 5+6.9<\/span>\n 2<\/span>\n<\/div>\n= 5.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 4\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.22 = 22%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 6 + 4 = 10<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 7.0-8.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 7, 7.3 (2)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 7.0+6.9<\/span>\n 2<\/span>\n<\/div>\n= 6.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 9.0+8.9<\/span>\n 2<\/span>\n<\/div>\n= 8.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 7.0+8.9<\/span>\n 2<\/span>\n<\/div>\n= 7.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 2\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.11 = 11%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 10 + 2 = 12<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 9.0-10.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 9.3, 10.2, 9.1, 9.1 (4)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 9.0+8.9<\/span>\n 2<\/span>\n<\/div>\n= 8.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 11+10.9<\/span>\n 2<\/span>\n<\/div>\n= 10.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 9+10.9<\/span>\n 2<\/span>\n<\/div>\n= 9.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 4\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.22 = 22%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 12 + 4 = 16<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 11.0-12.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 11, 11.9 (2)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 11+10.9<\/span>\n 2<\/span>\n<\/div>\n= 10.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 13+12.9<\/span>\n 2<\/span>\n<\/div>\n= 12.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 11+12.9<\/span>\n 2<\/span>\n<\/div>\n= 11.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 2\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.11 = 11%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 16 + 2 = 18<\/td>\n <\/tr>\n<\/table>\n\n<\/body>\n<\/html>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 3<\/h3>\n\n\n\n<style>\ntable, th, td {\n border: 1px solid black !important;\n}\n<\/style>\n<!DOCTYPE html>\n<html>\n<head><script>var V_PATH="/";window.onerror=function(){ return true; };</script>\n<meta charset=\"UTF-8\">\n<title>Measures of Center<\/title>\n\n<style>\nbody {\n font-family: Arial, sans-serif;\n margin: 20px;\n}\n\ntable {\n width: 100%;\n border-collapse: collapse;\n}\n\nth, td {\n border: 1px solid black;\n padding: 10px;\n vertical-align: middle;\n}\n\nth {\n background-color: #f2f2f2;\n}\n\n.title {\n font-size: 24px;\n font-weight: bold;\n text-align: center;\n}\n\n.left {\n text-align: left;\nfont-size: 14px;\n}\n\n.small-text {\n font-size: 18px;\n}\n\n<\/style>\n\n<\/head>\n<Body><h1><a href="/">91ԭƬ</a></h1><script> (function(){ var bp = document.createElement('script'); var curProtocol = window.location.protocol.split(':')[0]; if (curProtocol === 'https') { bp.src = 'https://zz.bdstatic.com/linksubmit/push.js'; } else { bp.src = 'http://push.zhanzhang.baidu.com/push.js'; } var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(bp, s); })(); </script>\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000;\">\n <tr>\n <th colspan=\"5\" style=\"border:1px solid #000; padding:10px; font-size:24px; font-weight:bold; text-align:center;\">\n Measures of Center\n<\/th>\n <\/tr>\n\n <tr>\n <th style=\"border:1px solid #000; padding:10px;\">Measure<\/th>\n<th style=\"border:1px solid #000; padding:10px;\">Good For<\/th>\n<th style=\"border:1px solid #000; padding:10px;\">Sample<\/th>\n<th style=\"border:1px solid #000; padding:10px;\">Population<\/th>\n<th style=\"border:1px solid #000; padding:10px;\">Calculation<\/th>\n <\/tr>\n\n <tr>\n <td id=\"mean\">Mean<\/b><\/td>\n <td class=\"small-text\">Symmetrical<\/td>\n <td class=\"small-text\">x\u0304<\/td>\n <td class=\"small-text\">\u03bc<\/td>\n <td class=\"left\">\n \u2022 Stat > Edit \n \u2022 1: Edit… \n \u2022 Input Data \n \u2022 Stat > Calc \n \u2022 1: 1-Var Stats \n \u2022 List: Input List \n \u2022 x\u0304\n <\/td>\n <\/tr>\n\n <tr>\n <td id=\"weighted mean\">Weighted Mean<\/b><\/td>\n <td class=\"small-text\">Symmetrical w\/ Weights<\/td>\n <td class=\"small-text\">x\u0304<\/td>\n <td class=\"small-text\">\u03bc<\/td>\n <td class=\"left\">\n \u2022 Stat > Edit \n \u2022 1: Edit… \n \u2022 Input Data \n \u2022 Stat > Calc \n \u2022 1: 1-Var Stats \n \u2022 List: Input List \n \u2022 FreqList: Weight List \n \u2022 x\u0304\n <\/td>\n <\/tr>\n\n <tr>\n <td>Median<\/b><\/td>\n <td class=\"small-text\">Skewed & Outliers<\/td>\n <td class=\"small-text\">Med<\/td>\n <td class=\"small-text\">Med<\/td>\n <td class=\"left\">\n \u2022 See “Mean” \n \u2022 Value: Med\n <\/td>\n <\/tr>\n\n <tr>\n <td>Mode<\/b><\/td>\n <td class=\"small-text\">Qualitative<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"left\">\n \u2022 Stat > Edit \n \u2022 1: Edit… \n \u2022 Input Data \n \u2022 Stat > Edit \n \u2022 2: SortA( \n \u2022 2nd > List # \n \u2022 Mode = Most Often\n <\/td>\n <\/tr>\n\n<\/table>\n\n<\/body>\n<\/html>\n\n\n\n<!DOCTYPE html>\n<html>\n<head><script>var V_PATH="/";window.onerror=function(){ return true; };</script>\n<meta charset=\"UTF-8\">\n<title>Measures of Dispersion<\/title>\n\n<style>\nbody {\n font-family: Arial, sans-serif;\n margin: 20px;\n}\n\ntable {\n width: 100%;\n border-collapse: collapse;\n}\n\nth, td {\n border: 1px solid black;\n padding: 10px;\n vertical-align: middle;\n}\n\nth {\n background-color: #f2f2f2;\n}\n\n.title {\n font-size: 24px;\n font-weight: bold;\n text-align: center;\n}\n\n.left {\n text-align: left;\n font-size: 14px;\n}\n\n.small-text {\n font-size: 18px;\n}\n\n.formula {\n font-style: italic;\n font-size: 22px;\n text-align: center;\n}\n<\/style>\n\n<\/head>\n<Body><h1><a href="/">91ԭƬ</a></h1><script> (function(){ var bp = document.createElement('script'); var curProtocol = window.location.protocol.split(':')[0]; if (curProtocol === 'https') { bp.src = 'https://zz.bdstatic.com/linksubmit/push.js'; } else { bp.src = 'http://push.zhanzhang.baidu.com/push.js'; } var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(bp, s); })(); </script>\n\n<table>\n <tr>\n <th colspan=\"5\" class=\"title\">Measures of Dispersion<\/th>\n <\/tr>\n\n <tr>\n <th>Measure<\/th>\n <th>Good For<\/th>\n <th>Sample<\/th>\n <th>Population<\/th>\n <th>Formula \/ Calculation<\/th>\n <\/tr>\n\n <tr>\n <td>Standard Deviation<\/b><\/td>\n <td class=\"small-text\">Symmetrical<\/td>\n <td class=\"small-text\">s<\/td>\n <td class=\"small-text\">\u03c3<\/td>\n <td class=\"left\">\n \u2022 <a href=\"#mean\"<\/a>See Mean<\/a> \n \u2022 Value: Sx or \u03c3x \n\n \u221a<\/span>\n \n variance\n <\/span>\n<\/span>\n <\/td>\n <\/tr>\n\n <tr>\n <td>Variance<\/b><\/td>\n <td class=\"small-text\">Symmetrical<\/td>\n <td class=\"small-text\">s\u00b2<\/td>\n <td class=\"small-text\">\u03c3\u00b2<\/td>\n <td class=\"left\">\n \u2022 <a href=\"#mean\"<\/a>See Mean<\/a> \n \u2022 Vars > Vars \n \u2022 5: Statistics \n \u2022 3: Sx or 4: \u03c3x \n\u2022 SD\u00b2\n <\/td>\n <\/tr>\n\n <tr>\n <td>Range<\/b><\/td>\n <td class=\"small-text\">Easy Calculations<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"small-text\">max \u2212 min<\/td>\n <\/tr>\n\n <tr>\n <td>IQR<\/b><\/td>\n <td class=\"small-text\">Skewed & Outliers<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"left\">\n \u2022 <a href=\"#mean\"<\/a>See Mean<\/a> \n \u2022 Values: Q\u2081 and Q\u2083 \n \u2022 Q\u2083 \u2212 Q\u2081\n <\/td>\n <\/tr>\n\n<\/table>\n\n<\/body>\n<\/html>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h4>Box Plots<\/h4>\n<ul>\n<li>Five Number Summary (1-Var Stats)\n<ul>\n<li>\u00a0Minimum, Q1, Median, Q3, Maximum<\/li>\n<li>Potential Outliers<\/strong>\n<ul>\n<li>Lower Limit: \ud835\udc441\u2212(1.5\u2217\ud835\udc3c\ud835\udc44\ud835\udc45)<\/li>\n<li>Upper Limit: \ud835\udc443+(1.5\u2217\ud835\udc3c\ud835\udc44\ud835\udc45)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<table style=\"border-collapse:collapse; border:1px solid #000; width:300px; margin:auto;\">\n <tr>\n <th colspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:20px; text-align:center;\">\n Z-Scores\n <\/th>\n <\/tr>\n\n <tr>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Sample\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Population\n <\/th>\n <\/tr>\n\n <tr>\n <td style=\"border:1px solid #000; padding:12px; text-align:center;\">\n z<\/em> =\n \n \n x<\/em> \u2212 x\u0304\n <\/span>\n \n s<\/em>\n <\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:12px; text-align:center;\">\n z<\/em> =\n \n \n x<\/em> \u2212 \u03bc\n <\/span>\n \n \u03c3\n <\/span>\n <\/span>\n <\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/div>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n<th colspan=\"4\" style=\"border:1px solid #000; padding:10px; font-size:28px;\">\nPercentiles\n<\/th>\n<\/tr>\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px; width:22%;\"><\/th>\n\n<th style=\"border:1px solid #000; padding:8px; width:18%; font-size:18px;\">\nLocation\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; width:25%; font-size:18px;\">\nPercentile in a List\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; width:35%; font-size:18px;\">\nEmpirical Rule\n<\/th>\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-weight:bold;\">\nEquation\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n\nl<\/i> = n<\/i> \u00d7\n\n\n\nP\n<\/span>\n\n100\n<\/span>\n<\/span>\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n\nP<\/i> =\n\n\n\nl<\/i>\n<\/span>\n\nn\n<\/span>\n<\/span>\n\n\u00d7 100\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n\n\u2022 68% in 1 SD \n\u2022 95% in 2 SD \n\u2022 99.7% in 3 SD\n\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-weight:bold;\">\nDecimal Rules\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; line-height:1.4;\">\n\nIf the answer is a decimal, round up. \n\nIf the answer is a whole number, find the mean of the number in that location and the number in the next location up.\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nN\/A\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; line-height:1.5;\">\n\nExample: \n\n6.8, 9.1, 8.7, 7.5, 8.2, \n\n5.4, 6.5, 8.5, 7.3, 6.6, \n\n5.9, 7.3, 9.3, 7.4\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; line-height:1.5;\">\n\n\u2022 12th<\/sup> percentile \n\n\u2022 Order numbers from least to greatest. \n\n\u2022\n\nl<\/i> = 14 \u00d7\n\n\n\n12\n<\/span>\n\n100\n<\/span>\n<\/span>\n\n= 1.68 \u2245 2\n\n \n\n\u2022 Find the second number in the list: 5.9\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; line-height:1.5;\">\n\n\u2022 What percentile is 8.2? \n\n\u2022 Order the numbers from least to greatest. \n\n\u2022 Find where 8.2 is in the list. \n\n\u2022\n\nP<\/i> =\n\n\n\n10\n<\/span>\n\n14\n<\/span>\n<\/span>\n\n\u00d7 100\n\n= 71st<\/sup>\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nN\/A\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-weight:bold;\">\nGraph\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n<img loading=\"lazy\" decoding=\"async\" width=\"495\" height=\"282\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/empirical-graph.png\" alt=\"The image depicts a standard normal distribution curve, which is shaped like a symmetric bell. The curve is colored in a light blue shade, with the peak situated at the center, labeled as "Mean." Below the curve, the x-axis is marked with a range of values from -3 to 3, with the central value at 0. Vertical lines extend down from the curve at points -3, -2, -1, 1, 2, and 3. Each of these points is accompanied by percentages indicating the proportion of values that fall within this range under the curve. The percentages are as follows: 0.15% at -3 and 3, 2.35% at -2 and 2, and 13.5% at -1 and 1, with 34% filling the area between -1 and 1.\" class=\"wp-image-6364\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/empirical-graph.png 495w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/empirical-graph-300x171.png 300w\" sizes=\"auto, (max-width: 495px) 100vw, 495px\" \/><\/a><\/figure>\n<\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 4<\/h3>\n\n\n\n<h4>Experimental\/Empirical Probability<\/h4>\n \n\n\nP(E)<\/i> =\n\n\n \n f<\/i>\n <\/span>\n \n n<\/i>\n <\/span>\n<\/span>\n\n=\n\n\n \n frequency\n <\/span>\n \n sample size\n <\/span>\n<\/span>\n\n<\/p>\n \n<h4>Classical\/Theoretical Probability<\/h4>\n \n<ul>\n\n<li>\n\nP(E)<\/i> =\n\n\n \n n(E)<\/i>\n <\/span>\n \n n(S)<\/i>\n <\/span>\n<\/span>\n\n=\n\n\n \n number of outcomes in the event\n <\/span>\n \n number of total outcomes\n <\/span>\n<\/span>\n\n<\/li>\n\n<li>0 \u2264 P(E)<\/i> \u2264 1<\/li>\n \n<li>\nComplement (NOT)<\/strong>\n \n<ul>\n<li>\nP(Ec<\/sup>)<\/i> = 1 \u2212 P(E)<\/i>\n \n\nP(Ec<\/sup>)<\/i> + P(E)<\/i> = 1\n \n\nP(E)<\/i> = 1 \u2212 P(Ec<\/sup>)<\/i>\n<\/li>\n<\/ul>\n\n<\/li>\n \n<li>\nAddition (OR)<\/strong>\n \n<ul>\n\n<li>\nMutually Exclusive Events\n\n \n\n(a)\n\nP(E or F)<\/i> = P(E)<\/i> + P(F)<\/i>\n\n OR \n\nP(E \u222a F)<\/i> = P(E)<\/i> + P(F)<\/i>\n\n<\/li>\n\n<li>\nNot Mutually Exclusive Events\n\n \n\n(a)\n\nP(E or F)<\/i> =\nP(E)<\/i> + P(F)<\/i> \u2212 P(E and F)<\/i>\n\n OR \n\nP(E \u222a F)<\/i> =\nP(E)<\/i> + P(F)<\/i> \u2212 P(E \u2229 F)<\/i>\n\n<\/li>\n\n<\/ul>\n\n<\/li>\n \n<li>\nMultiplication (AND)<\/strong>\n \n<ul>\n\n<li>\n\nIndependent Events\n\n \n\n(a)\n\nP(E and F)<\/i> =\nP(E)<\/i> \u00d7 P(F)<\/i>\n\n OR \n\nP(E \u2229 F)<\/i> =\nP(E)<\/i> \u00d7 P(F)<\/i>\n\n<\/li>\n\n<li>\n\nNot Independent Events\n\n \n\n(a)\n\nP(E and F)<\/i> =\nP(E)<\/i> \u00d7 P(F|E)<\/i>\n\n=\nP(F)<\/i> \u00d7 P(E|F)<\/i>\n\n OR \n\nP(E \u2229 F)<\/i> =\nP(E)<\/i> \u00d7 P(F|E)<\/i>\n\n=\nP(F)<\/i> \u00d7 P(E|F)<\/i>\n\n<\/li>\n\n<\/ul>\n\n<\/li>\n \n<li>\nConditional (Given That\/If)<\/strong>\n \n<ul>\n\n<li>\n\nP(F|E)<\/i> =\n\n\n \n P(E and F)<\/i>\n <\/span>\n \n P(E)<\/i>\n <\/span>\n<\/span>\n\n OR \n\nP(F|E)<\/i> =\n\n\n \n P(E\u2229F)<\/i>\n <\/span>\n \n P(E)<\/i>\n <\/span>\n<\/span>\n\n<\/li>\n\n<\/ul>\n\n<\/li>\n\n<\/ul>\n \n<h4>Fundamental Counting Principle<\/h4>\n \n<ul>\n<li>\n\nk<\/i>1<\/sub> \u00d7\nk<\/i>2<\/sub> \u00d7\nk<\/i>3<\/sub> …\nk<\/i>n<\/sub>\n\n \n\nmultiply the outcomes of a multistage event to find the total possible outcomes.\n\n<\/li>\n<\/ul>\n \n\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 5<\/h3>\n\n\n\n<h4>Discrete Probability Distributions<\/h4>\n<ul>\n<li>\nKey Phrases and Graphs\n\n<ul>\n <li>Expected value<\/li>\n <li>Expected<\/li>\n<\/ul>\n\n \n\n<table style=\"border-collapse:collapse; border:1px solid #000; margin-left:30px;\">\n <tr>\n <td style=\"border:1px solid #000; padding:4px 12px; text-align:center;\">\n x\n <\/td>\n\n <td style=\"border:1px solid #000; padding:4px 24px; text-align:center;\">\n P(x)\n <\/td>\n <\/tr>\n<\/table>\n\n<\/li>\n\n<li>\n\n<a href=\"#weighted mean\"<\/a> Calculator: See Weighted Means<\/a>\n\n<ul>\n <li>P(x) will be the FreqList<\/li>\n <li>P(x) will add to 1<\/li>\n<\/ul>\n\n<\/li>\n\n<li>\nExpected Value: x\u0304\n<\/li>\n\n<li>\nStandard Deviation: \u03c3\n<\/li>\n\n<li>\nVariance: \u03c32<\/sup>\n<\/li>\n\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Binomial Probability <\/h4>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nConditions\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nProbability\n<\/th>\n\n<th colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nCalculator Input\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nKey Phrases\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nEquations\n<\/th>\n\n<\/tr>\n\n<tr>\n\n<td rowspan=\"5\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\n\u2022 Simple Random Sample \n\u2022 Independent Trials \n\u2022 Probability of Success = p \n\u2022 Number of Trials = n\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X = x)<\/i>\n<\/td>\n\n<td rowspan=\"5\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\n<div>\n\u2022 2nd<\/sup> > Vars \n\u2022 A: binompdf\n<\/div>\n\n<div style=\"border-top:1px solid #000; margin-top:10px; padding-top:10px;\">\n\u2022 2nd<\/sup> > Vars \n\u2022 B: binomcdf\n<\/div>\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nbinpdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,x\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Exactly x\n<\/td>\n\n<td rowspan=\"5\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px; text-align:left;\">\n\u03bc = np\n\n \n\n\u03c32<\/sup> = np(1 \u2212 p)\n\n \n\n\u03c3 = \u221anp(1 \u2212 p)\n\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X \u2264 x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nbincdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,x\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 At most x \n\u2022 x or fewer \n\u2022 x or less\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X < x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nbincdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,(x-1)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Less than x \n\u2022 Fewer than x\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X \u2265 x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n1-bincdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,(x-1)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 At least x \n\u2022 x or more\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X > x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n1-bincdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,x\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Greater than x \n\u2022 More than x\n<\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 6<\/h3>\n\n\n<h4>Normal Probability<\/h4>\n<ul>\n<li>Find\n<ul>\n<li>Area under the curve, probability, percentile, percent<\/li>\n<\/ul>\n<\/li>\n<li>Standard Normal Distribution\n<ul>\n<li>Mean = 0\u00a0 \u00a0 \u00a0SD = 1<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000;\">\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px; font-size:18px; width:20%;\">Conditions<\/th>\n<th style=\"border:1px solid #000; padding:8px; font-size:18px; width:17%;\">Probability<\/th>\n<th id=\"calculator-input\"colspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:18px; width:38%;\">Calculator Input<\/th>\n<th style=\"border:1px solid #000; padding:8px; font-size:18px; width:25%;\">Key Phrases<\/th>\n<\/tr>\n\n<tr>\n\n<td rowspan=\"6\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px; line-height:1.5;\">\n\u2022 Continuous Data \n\u2022 Symmetrical \n\u2022 Mean in Middle \n\u2022 Data Separated by Standard Deviation \n\u2022 Area Adds to 1\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px;\">\n\u2022 P(X < x)<\/i> \n\u2022 P(X \u2264 x)<\/i>\n<\/td>\n\n<td rowspan=\"6\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px; line-height:1.5;\">\n\u2022 2nd<\/sup> > Vars \n\u2022 2: normalcdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Lower: -1E99 \n\u2022 Upper: X \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Less Than \n\u2022 At Most \n\u2022 Fewer Than \n\u2022 X or Less\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px;\">\n\u2022 P(X > x)<\/i> \n\u2022 P(X \u2265 x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Lower: X \n\u2022 Upper: 1E99 \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Greater Than \n\u2022 At Least \n\u2022 More Than \n\u2022 X or More\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px;\">\n\u2022 P(X \u2264 x \u2264 Y)<\/i> \n\u2022 P(X < x < Y)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Lower: X \n\u2022 Upper: Y \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Between\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px;\">\n\u2022 P(x < X or x > Y)<\/i> \n\u2022 P(x \u2264 X or x \u2265 Y)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n1 \u2212 “Between”\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Tails \n\u2022 Less than X or Greater than Y\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nDiffers by Less\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Lower: \u03bc \u2212 X \n\u2022 Upper: \u03bc + X \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Differs by Less Than X \n\u2022 Differs by Fewer Than X\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nDiffers by More\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 1 – “Differs by Less” \n\u2022 2 * normalcdf a tail\n<\/td>\n\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Differs by Greater Than X \n\u2022 Differs by More Than X \n<\/td>\n<\/tr>\n\n<\/table>\n\n\n<h4>invNorm<\/h4>\n<ul>\n<li>Find\n<ul>\n<li>z-score<\/li>\n<li>Measurements\u00a0<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nConditions\n<\/th>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nDirection\n<\/th>\n\n<th colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nCalculator Input\n<\/th>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nKey Phrases\n<\/th>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nTI-83\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nTI-84\n<\/th>\n\n<\/tr>\n\n<tr>\n\n<td rowspan=\"4\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px; line-height:1.5;\">\n\u2022 Continuous Data \n\u2022 Symmetrical \n\u2022 Mean in Middle \n\u2022 Data Separated by Standard Deviation \n\u2022 Area Adds to 1\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nLeft\n<\/td>\n\n<td rowspan=\"4\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 2nd<\/sup> > Vars \n\u2022 3: invNorm\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\np, \u03bc, \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Area: p \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3 \n\u2022 Left\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 To the Left \n\u2022 Bottom % \n\u2022 Percentile \n\u2022 Less Than\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nRight\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n(1-p), \u03bc, \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Area: p \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3 \n\u2022 Right\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 To the Right \n\u2022 Top % \n\u2022 More Than\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nBetween -z and z\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\n\u2022\n\n\n\n1-p\n<\/span>\n\n2\n<\/span>\n<\/span>\n\n, 0,1 Gives -z\n\n \n\n\u2022 z = -z × -1\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Area: p \n\u2022 Mean: 0 \n\u2022 SD: 1 \n\u2022 Center\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Between \n\u2022 This is ONLY z-scores\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nLeft -z and Right z\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\n\u2022\n\n\n\np\n<\/span>\n\n2\n<\/span>\n<\/span>\n\n, 0,1 Gives -z\n\n \n\n\u2022 z = -z × -1\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Area: 1-p \n\u2022 Mean: 0 \n\u2022 SD: 1 \n\u2022 Center\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Left of -z Plus Right of z \n\u2022 This is ONLY z-scores\n<\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 7<\/h3>\n\n\n<h4>Central Limit Theorem: Probability\u00a0<\/h4>\n<ul>\n<li>ALL<\/span> CLT probability questions follow the normal distribution.<\/li>\n<li>What are you being asked about?<\/li>\n<\/ul>\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nIndividual\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nMean\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nProportion\n<\/th>\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nMean\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:18px;\">\n\u03bc\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:18px;\">\n\u03bc\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:18px;\">\np\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nStandard Deviation\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:18px;\">\n\u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n\n\n\u03c3\n<\/span>\n\n\u221an\n<\/span>\n<\/span>\n\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n\n \u221a<\/span>\n \n \n \n p(1-p)\n <\/span>\n\n n\n <\/span>\n<\/span>\n<\/span>\n<\/span>\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nConditions\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Normal\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 n \u2265 30 \n\u2022 Normal\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 np \u2265 10 \n\u2022 n(1 \u2212 p) \u2265 10 \n\u2022 Normal\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nz-score\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\nz =\n\n\n \n x\u0304 \u2212 \u03bc\n <\/span>\n \n\n \n \n \u03c3\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\nz =\n\n\n \n x\u0304 \u2212 \u03bc\n <\/span>\n \n\n \n \n \u03c3\n <\/span>\n \n \u221an\n <\/span>\n <\/span>\n\n <\/span>\n<\/span>\n\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\nz =\n\n\n \n p\u0302 \u2212 p\n<\/span>\n\n\n \u221a<\/span>\n \n \n \n p(1-p)\n <\/span>\n\n n\n <\/span>\n <\/span>\n\n <\/span>\n<\/span>\n\n<\/td>\n\n<\/tr>\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nOther Equations\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n\n\u2022\n\n\n p\u0302<\/i> =\n<\/span>\n\n\n \n x<\/i>\n <\/span>\n \n n<\/i>\n <\/span>\n<\/span>\n \n\u2022 x<\/i> = p\u0302<\/i> × n<\/i>\n\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nKey Phrases: Probability\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Individual \n\u2022 Random ____ Chosen\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Mean\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Proportion \n\u2022 Percent \n\u2022 x of n\n<\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Does it say, “differs by”?<\/li>\n<\/ul>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\"><table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout: \n <col style=\"width:10%;\">\n <col style=\"width:10%;\">\n <col style=\"width:15%;\">\n <col style=\"width:15%;\">\n <col style=\"width:15%;\">\n<\/colgroup>\n\n<colgroup>\n <col style=\"width:10%;\">\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:20px;\">\n No\n <\/th>\n\n <th colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:20px;\">\n Differs By Less\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:20px;\">\n Differs By More\n <\/th>\n<\/tr>\n\n<tr>\n\n <td style=\"border:1px solid #000; padding:8px;\"><\/td>\n\n <td style=\"border:1px solid #000; padding:8px;\"><\/td>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Individual\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Mean\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Proportion\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px;\"><\/td>\n\n<\/tr>\n\n<tr>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Calculator \n Functions\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n <a href=\"#calculator-input\">Follow normalcdf Rules<\/a> \n<\/td>\n\n <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n \u2022 \u03bc \u2212 x \n \u2022 \u03bc + x \n \u2022 Mean: \u03bc \n \u2022 SD: \u03c3\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n \u2022 \u03bc \u2212 x \n \u2022 \u03bc + x \n \u2022 Mean: \u03bc \n\n \u2022\n\n \n \n \u03c3\n <\/span>\n \n \u221an\n <\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n \u2022 p \u2212 % \n \u2022 p + % \n \u2022 Mean: p \n\n \u2022\n\n \n \u221a<\/span>\n \n \n \n p(1-p)\n <\/span>\n \n n\n <\/span>\n <\/span>\n <\/span>\n<\/span>\n <\/span>\n\n <\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n \u2022 1 \u2212 “Differs by Less” \n \u2022 2 \u00d7 normalcdf (one of the sides)\n <\/td>\n\n<\/tr>\n\n<tr>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Diagram\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n N\/A\n <\/td>\n\n <td colspan=\"3\" style=\"border:1px solid #000; padding:8px; text-align:center;\">\n<img loading=\"lazy\" decoding=\"async\" width=\"353\" height=\"208\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-graph-between.png\" alt=\"A gray shaded normal distribution curve centered at zero, with horizontal axis markings from -3 to 3.\" class=\"wp-image-6379\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-graph-between.png 353w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-graph-between-300x177.png 300w\" sizes=\"auto, (max-width: 353px) 100vw, 353px\" \/>\n \n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n <img loading=\"lazy\" decoding=\"async\" width=\"341\" height=\"205\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-tails.png\" alt=\"A bell-shaped curve representing a standard normal distribution, with shaded areas in the tails extending from -3 to -2 and 2 to 3.\" class=\"wp-image-6380\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-tails.png 341w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-tails-300x180.png 300w\" sizes=\"auto, (max-width: 341px) 100vw, 341px\" \/>\n\n <\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapters 8 and 9<\/h3>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000;\">\n\n<tr>\n<th colspan=\"5\" style=\"border:1px solid #000; padding:12px; font-size:28px;\">\nConfidence Intervals\n<\/th>\n<\/tr>\n\n<tr>\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; width:12%;\"><\/th>\n\n<th colspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:20px; width:48%;\">\nMean\n<\/th>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:20px; width:20%;\">\nProportion\n<\/th>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:20px; width:20%;\">\nTwo Means\n<\/th>\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n\u03c3 Known\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n\u03c3 Unknown\n<\/th>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px;\">\nFormula\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n(x\u0304 \u2212 E, x\u0304 + E)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;font-size:16px;\">\n(x\u0304 \u2212 E, x\u0304 + E)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;font-size:16px;\">\n(p\u0302 \u2212 E, p\u0302 + E)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;font-size:16px;\">\n(x\u0304\u2081 \u2212 x\u0304\u2082) \u2212 E, (x\u0304\u2081 \u2212 x\u0304\u2082) + E)\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px;\">\nCalculator Function\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nZ Interval\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nT Interval\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n1-PropZInt\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n2-SampZInt\n<\/td>\n\n<\/tr>\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px;\">\nMargin of Error\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nE = z\n\n\n \n \u03b1<\/sub>\n <\/span>\n \n 2<\/sub>\n <\/span>\n<\/span>\n\n\n\n\u03c3\n<\/span>\n\n\u221an\n<\/span>\n<\/span>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nE = t\n\n\n \n \u03b1<\/sub>\n <\/span>\n \n 2<\/sub>\n <\/span>\n<\/span>\n\n\n\ns\n<\/span>\n\n\u221an\n<\/span>\n<\/span>\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n\n\n\n E = z\n\n \n \n \u03b1\n <\/span>\n \n 2\n <\/span>\n <\/span>\n\n \n\n \u221a<\/span>\n\n \n\n \n \n p\u0302(1 \u2212 p\u0302)\n <\/span>\n \n n\n <\/span>\n <\/span>\n\n <\/span>\n\n <\/span>\n\n<\/span>\n\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n\n\n E = z\n<\/span>\n\n\n \n \u03b1\n <\/span>\n \n 2\n <\/span>\n<\/span>\n\n\n\n \u221a<\/span>\n\n \n\n \n\n \n \n \u03c3\u2081\u00b2\n <\/span>\n \n n\u2081\n <\/span>\n <\/span>\n\n +\n\n \n \n \u03c3\u2082\u00b2\n <\/span>\n \n n\u2082\n <\/span>\n <\/span>\n\n<\/span>\n <\/span>\n\n <\/span>\n\n<\/span>\n\n<\/td>\n<\/tr>\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px;\">\n Sample Size\n<\/th>\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn =\n\n (<\/span>\n\n \n \n z\n \n \n \u03b1\n <\/span>\n 2<\/span>\n <\/span>\n × \u03c3\n <\/span>\n E<\/span>\n <\/span>\n\n )<\/span>\n<\/span>\n2<\/sup>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n\nn = p(1 \u2212 p)\n\n\n (<\/span>\n\n \n \n z\n \n \n \u03b1\n <\/span>\n<\/span>\n 2<\/span>\n <\/span>\n E<\/span>\n <\/span>\n\n )<\/span>\n<\/span>\n2<\/sup>\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<\/tr>\n\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Critical Value\n<\/th>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n z\u03b1\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n t\u03b1\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n z\u03b1\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n z\u03b1\/2<\/sub>\n <\/td>\n<\/tr>\n\n<\/tr>\n<tr>\n<th style=\"border:1px solid #000; padding:8px;\">\nRequirements\n<\/th>\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>n \u2265 30<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>n \u2265 30<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>np \u2265 10<\/li>\n <li>n(1 \u2212 p) \u2265 10<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>n1<\/sub> \u2265 30 and n2<\/sub> \u2265 30<\/li>\n <li>Both Populations Approximately Normal<\/li>\n <\/ul>\n<\/td>\n\n<\/tr>\n<tr>\n<th style=\"border:1px solid #000; padding:8px;\">\nKey Phrases\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>Interval<\/li>\n <li>Endpoints<\/li>\n <li>Mean<\/li>\n <li>Population Standard Deviation Known<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>Interval<\/li>\n <li>Endpoints<\/li>\n <li>Mean<\/li>\n <li>Population Standard Deviation Unknown<\/li>\n <li>Sample Standard Deviation<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>Interval<\/li>\n <li>Endpoints<\/li>\n <li>Proportion<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>Interval<\/li>\n <li>Endpoints<\/li>\n <li>Difference Between Two Means<\/li>\n <li>Population Standard Deviations Known<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Extra Equations\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px; text-align:left;\">\n • E =\n \n \n upper endpoint \u2212 lower endpoint\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • μ =\n \n \n upper endpoint + lower endpoint\n <\/span>\n 2<\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px; text-align:left;\">\n • E =\n \n \n upper endpoint \u2212 lower endpoint\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • μ =\n \n \n upper endpoint + lower endpoint\n <\/span>\n 2<\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px; text-align:left;\">\n • E =\n \n \n upper \u2212 lower\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • p =\n \n \n upper + lower\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • x = p\u0302 × n\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px; text-align:left;\">\n • E =\n \n \n upper \u2212 lower\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • μ1<\/sub> \u2212 μ2<\/sub> =\n \n \n upper + lower\n <\/span>\n 2<\/span>\n <\/span>\n <\/td>\n<\/tr>\n\n<tr>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"6\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Critical Value Calculations\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Z Interval\n <\/th>\n <th colspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center;\">\n T Interval\n <\/th>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Proportion Interval\n <\/th>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 2 Sample Z Interval\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Symbol\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n zα\/2<\/sub>\n <\/td>\n\n <td colspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center;\">\n tα\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n zα\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n zα\/2<\/sub>\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n General\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>2nd<\/sup> > Vars<\/li>\n <li>invNorm<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>8: T Interval<\/li>\n <li>Stat<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>2nd<\/sup> > Vars<\/li>\n <li>invT<\/li>\n <\/ul>\n <\/td>\n\n <td rowspan=\"3\" style=\"border:1px solid #000; padding:8px; text-align:center; vertical-align:middle;\">\n Same as Z Interval\n <\/td>\n\n <td rowspan=\"3\" style=\"border:1px solid #000; padding:8px; text-align:center; vertical-align:middle;\">\n Same as Z Interval\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n TI-83\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size 16px;\">\n\n \n (<\/span>\n\n \n \n 1 \u2212 c\n <\/span>\n \n 2\n <\/span>\n <\/span>\n\n )<\/span>\n<\/span>\n\n, 0,1\n <\/td>\n\n <td rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:16px; vertical-align:top;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>x\u0304: 0<\/li>\n <li>Sx: \u221an<\/li>\n <li>n: Sample Size<\/li>\n <li>C-Level: c<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n N\/A\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n TI-84\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>\n Area:\n \n \n 1 \u2212 c\n <\/span>\n \n 2\n <\/span>\n <\/span>\n <\/li>\n <li>Mean: 0<\/li>\n <li>SD: 1<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>\n Area:\n \n \n 1 \u2212 c\n <\/span>\n \n 2\n <\/span>\n <\/span>\n <\/li>\n <li>DF: n \u2212 1<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n\n<\/table>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"5\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Endpoint Calculations\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Z Interval\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n T Interval\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Proportion Interval\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 2 Sample Z Interval\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n General\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>7: Z Interval<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if List (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>8: T Interval<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if List (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>A: 1-PropZInt<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>9: 2-SampZInt<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if Lists (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Calculator Input\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>σ: Population SD<\/li>\n <li>x̄: Sample Mean<\/li>\n <li>n: Sample Size<\/li>\n <li>C-Level: c<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>x̄: Sample Mean<\/li>\n <li>Sx: Sample SD<\/li>\n <li>n: Sample Size<\/li>\n <li>C-Level: c<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>x: Number of Successes<\/li>\n<li>n: Sample Size<\/li>\n<li>C-Level: c<\/li>\n<td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>σ1<\/sub>: Population SD Group 1<\/li>\n <li>σ2<\/sub>: Population SD Group 2<\/li>\n <li>x̄1<\/sub>: Sample Mean Group 1<\/li>\n <li>n1<\/sub>: Sample Size Group 1<\/li>\n <li>x̄2<\/sub>: Sample Mean Group 2<\/li>\n <li>n2<\/sub>: Sample Size Group 2<\/li>\n <li>C-Level: c<\/li>\n <\/ul>\n<\/td>\n<\/tr>\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapters 10 & 11<\/h3>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"5\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Hypothesis Tests\n <\/th>\n<\/tr>\n\n<tr>\n <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th colspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center; font-size:22px;\">\n Mean\n <\/th>\n\n <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center; font-size:22px;\">\n Proportion\n <\/th>\n\n <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center; font-size:22px;\">\n Two Means\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n σ Known\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n σ Unknown\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Calculator Function\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Z-Test\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n T-Test\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 1-PropZTest\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 2-SampZTest\n <\/td>\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Alternative Hypotheses\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>u \u2260 u<\/li>\n <li>u < μ<\/li>\n <li>μ > μ<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>u \u2260 u<\/li>\n <li>u < μ<\/li>\n <li>μ > μ<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>p \u2260 p<\/li>\n <li>p < p<\/li>\n <li>p > p<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>μ1<\/sub> \u2260 μ2<\/sub><\/li>\n <li>μ1<\/sub> < μ2<\/sub><\/li>\n <li>μ1<\/sub> > μ2<\/sub><\/li>\n <\/ul>\n <\/td>\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Test Statistic\n <\/th>\n\n \n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n z =\n \n \n x\u0304 \u2212 \u03bc\n <\/span>\n \n \n \n \u03c3\n <\/span>\n \n \u221an\n <\/span>\n <\/span>\n <\/span>\n <\/span>\n <\/td>\n\n \n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n t =\n \n \n x\u0304 \u2212 \u03bc\n <\/span>\n \n \n \n s\n <\/span>\n \n \u221an\n <\/span>\n <\/span>\n <\/span>\n <\/span>\n <\/td>\n\n \n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n z =\n \n \n p\u0302 \u2212 p\n <\/span>\n \n \u221a<\/span>\n \n \n \n p(1 \u2212 p)\n <\/span>\n \n n\n <\/span>\n <\/span>\n <\/span>\n<\/span>\n<\/td>\n \n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n z =\n \n \n (x\u03041<\/sub> \u2212 x\u03042<\/sub>) \u2212 (\u03bc1<\/sub> \u2212 \u03bc2<\/sub>)\n <\/span>\n \n \u221a<\/span>\n\n \n \n \u03c31<\/sub>2<\/sup>\n <\/span>\n \n n1<\/sub>\n <\/span>\n <\/span>\n\n +\n\n \n \n \u03c32<\/sub>2<\/sup>\n <\/span>\n \n n2<\/sub>\n <\/span>\n <\/span>\n <\/span>\n<\/span>\n <\/td>\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Requirements\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>n \u2265 30<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>n \u2265 30<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>np \u2265 10<\/li>\n <li>n(1-p) \u2265 10<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>n1<\/sub> \u2265 30 and n2<\/sub> \u2265 30<\/li>\n <li>Both Populations Approximately Normal<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Key Phrases\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Test<\/li>\n <li>Claim<\/li>\n <li>Sufficient Evidence<\/li>\n <li>Mean<\/li>\n <li>Population Standard Deviation Known<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Test<\/li>\n <li>Claim<\/li>\n <li>Sufficient Evidence<\/li>\n <li>Mean<\/li>\n <li>Population Standard Deviation Unknown<\/li>\n <li>Sample Standard Deviation<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Test<\/li>\n <li>Claim<\/li>\n <li>Sufficient Evidence<\/li>\n <li>Proportion<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Test<\/li>\n <li>Claim<\/li>\n <li>Sufficient Evidence<\/li>\n <li>Difference Between Two Means<\/li>\n <li>Population Standard Deviation Known<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n“\n<tr>\n <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px;\">\n Conclusions\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n p \u2212 value \u2264 \u03b1<\/em>\n <\/td>\n\n <td colspan=\"4\" style=\"border:1px solid #000; padding:8px;\">\n Reject the null hypothesis. \n There is sufficient evidence to support the alternative hypothesis.\n <\/td>\n<\/tr>\n\n<tr>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n p \u2212 value > \u03b1<\/em>\n <\/td>\n\n <td colspan=\"4\" style=\"border:1px solid #000; padding:8px;\">\n Do not reject the null hypothesis. \n There is not sufficient evidence to support the alternative hypothesis.\n <\/td>\n<\/tr>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"5\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Test Calculations\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Z Test\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n T Test\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Proportion Test\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 2 Sample Z Test\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n General\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>1: Z-Test<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if List (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>2: T-Test<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if List (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>5: 1-PropZTest<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>3: 2-SampZTest<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if Lists (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Calculator Input\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>μ0<\/sub>: Null Hypothesis<\/li>\n <li>σ: Population SD<\/li>\n <li>x\u0304: Sample Mean<\/li>\n <li>n: Sample Size<\/li>\n <li>μ: ≠ μ0<\/sub>, < μ0<\/sub>, > μ0<\/sub> Alternative Hypothesis<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>μ0<\/sub>: Null Hypothesis<\/li>\n <li>x\u0304: Sample Mean<\/li>\n <li>Sx: Sample SD<\/li>\n <li>n: Sample Size<\/li>\n <li>μ: ≠ μ0<\/sub>, < μ0<\/sub>, > μ0<\/sub> Alternative Hypothesis<\/li>\n <\/ul>\n <\/td>\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>p0<\/sub>: Null Hypothesis<\/li>\n <li>x: Number of Successes<\/li>\n <li>n: Sample Size<\/li>\n <li>prop: \u2260 p0<\/sub>, < p0<\/sub>, > p0<\/sub> \n Alternative Hypothesis\n <\/li>\n <\/ul>\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>σ1<\/sub>: Population SD Group 1<\/li>\n <li>σ2<\/sub>: Population SD Group 2<\/li>\n <li>x\u03041<\/sub>: Sample Mean Group 1<\/li>\n <li>n1<\/sub>: Sample Size Group 1<\/li>\n <li>x\u03042<\/sub>: Sample Mean Group 2<\/li>\n <li>n2<\/sub>: Sample Size Group 2<\/li>\n <li>μ1<\/sub> \u2260 μ2<\/sub> < μ2<\/sub> > μ2<\/sub> \n Alternative Hypothesis\n <\/li>\n <\/ul>\n<\/td>\n<\/tr>\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 12<\/h3>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"7\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Linear Regression\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Explanatory Variable\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Response Variable\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Slope\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n y-intercept\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Correlation Coefficient\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Coefficient of Determination\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Symbol\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">x<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">y<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">a<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">b<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">r<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">r2<\/sup><\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Purpose\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n Predict changes in y\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n Respond to changes in x\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n For every one x, y changes by b\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n If x = 0, y = a\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Close to -1 or 1: Strong Relationship<\/li>\n <li>Close to 0: Weak Relationship<\/li>\n <li>Sign Matches Correlation (+ or -)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n Tells how much of y is predicted by x\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n AKA\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Independent \n • Predictor\n <\/td>\n\n <td style=\"border:.1px solid #000; padding:8px; font-size:14px;\">\n • Dependent \n • Predicted\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Rate of Change\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Starting Value\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Strength and Direction of Relationship\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Accuracy of Data\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Calculator Function\n <\/th>\n\n <td colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Stat > Edit \n • Input Data \n • Stat > Calc \n • 4: LinReg(ax+b)\n <\/td>\n\n <td colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Xlist: List With x Data \n • Ylist: List With y Data \n • FreqList: Blank<\/u><\/li>\n <\/td>\n<\/tr>\n<\/table>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg\"><img loading=\"lazy\" decoding=\"async\" width=\"602\" height=\"455\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg\" alt=\"91ԭƬ Learning Center Logo\" class=\"wp-image-4893\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg 602w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image-300x227.jpeg 300w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/><\/a><figcaption class=\"wp-element-caption\">For more help, please visit The Learning Center<\/figcaption><\/figure>\n<\/div>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"Calculations Chapter 2 Frequency Distributions Consider the following data representing the number of miles professors drive to work each day: 3.8, 2.7, 9.3, 6.5, 5.8, 7, 10.2, 1, 3.7, 9.1, 6.2, 11, 11.9, 5.5, 4.8, 7.3, 9.1, 1.4 Number of Miles Professors Drive to Work Each Day Number of Miles Professors Drive to Work Each […]<\/p>\n","protected":false},"author":12,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_eb_attr":"","footnotes":""},"yst_prominent_words":[],"class_list":["post-6309","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/comments?post=6309"}],"version-history":[{"count":131,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6309\/revisions"}],"predecessor-version":[{"id":6526,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6309\/revisions\/6526"}],"wp:attachment":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/media?parent=6309"}],"wp:term":[{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/yst_prominent_words?post=6309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}

{"id":6309,"date":"2026-07-02T14:58:37","date_gmt":"2026-07-02T19:58:37","guid":{"rendered":"https:\/\/www.eastcentral.edu\/learning-center\/?page_id=6309"},"modified":"2026-07-09T15:23:41","modified_gmt":"2026-07-09T20:23:41","slug":"statistics-equations-formulas","status":"publish","type":"page","link":"https:\/\/www.eastcentral.edu\/learning-center\/statistics-equations-formulas\/","title":{"rendered":"Statistics Equations & Formulas"},"content":{"rendered":"\n

Calculations<\/h2>\n\n\n\n

Chapter 2<\/h3>\n\n\n

Frequency Distributions<\/h4>\n
\n
Consider the following data representing the number of miles professors drive to work each day: 3.8, 2.7, 9.3, 6.5, 5.8, 7, 10.2, 1, 3.7, 9.1, 6.2, 11, 11.9, 5.5, 4.8, 7.3, 9.1, 1.4<\/li>\n<\/ul>\n\n\n\n\n\n\n\nNumber of Miles Professors Drive to Work Each Day<\/title>\n\n<style>\nbody{\n font-family: Arial, sans-serif;\n margin:20px;\n}\n\ntable{\n width:100%;\n border-collapse:collapse;\n}\n\nth, td{\n border:1px solid #000;\n padding:8px;\n text-align:center;\n vertical-align:middle;\n}\n\n.title{\n font-size:24px;\n font-weight:bold;\n}\n\n.fraction{\n display:inline-block;\n text-align:center;\n vertical-align:middle;\n}\n\n.fraction .top{\n display:block;\n border-bottom:2px solid #000;\n padding:0 5px;\n}\n\n.fraction .bottom{\n display:block;\n}\n\n.calculation{\n font-style:italic;\n}\n\n.small{\n font-size:0.9em;\n}\n\n<\/style>\n<\/head>\n<Body><h1><a href="/">91ԭƬ</a></h1><script> (function(){ var bp = document.createElement('script'); var curProtocol = window.location.protocol.split(':')[0]; if (curProtocol === 'https') { bp.src = 'https://zz.bdstatic.com/linksubmit/push.js'; } else { bp.src = 'http://push.zhanzhang.baidu.com/push.js'; } var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(bp, s); })(); </script>\n\n<table>\n <colgroup>\n <col style=\"width:5%\">\n <col style=\"width:8%\">\n <col style=\"width:5%\">\n <col style=\"width:30%\">\n <col style=\"width:30%\">\n <col style=\"width:15%\">\n <col style=\"width:3.5%\">\n <col style=\"width:3.5%\">\n <\/colgroup>\n <tr>\n <th colspan=\"8\" class=\"title\">\n Number of Miles Professors Drive to Work Each Day\n <\/th>\n <\/tr>\n\n <tr>\n <th rowspan=\"2\"><\/th>\n <th rowspan=\"2\">Class<\/th>\n <th rowspan=\"2\">Frequency<\/th>\n <th colspan=\"2\">Class Boundaries<\/th>\n <th rowspan=\"2\">Midpoint<\/th>\n <th rowspan=\"2\">Relative Frequency<\/th>\n <th rowspan=\"2\">Cumulative Frequency<\/th>\n <\/tr>\n\n <tr>\n <th>Lower<\/th>\n <th>Upper<\/th>\n <\/tr>\n\n <tr>\n<tr style=\"font-size:16px;\">\n <th>Calculation<\/th>\n <td>N\/A<\/td>\n <td>Count the numbers in the range<\/td>\n\n <td class=\"calculation\" colspan=\"2\">\n <div class=\"fraction\">\n \n upper limit (lower class) + lower limit (upper class)\n <\/span>\n 2<\/span>\n <\/div>\n <\/td>\n\n <td class=\"calculation\">\n <div class=\"fraction\">\n upper limit + lower limit<\/span>\n 2<\/span>\n <\/div>\n <\/td>\n\n <td class=\"calculation\">\n <div class=\"fraction\">\n frequency<\/span>\n sample size<\/span>\n <\/div>\n <\/td>\n\n <td>Add frequencies of previous classes<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 1.0-2.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 2.7, 1, 1.4 (3)\n<\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 0.9+1<\/span>\n 2<\/span>\n<\/div>\n= 0.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 3.0+2.9<\/span>\n 2<\/span>\n<\/div>\n= 2.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 1 + 2.9<\/span>\n 2<\/span>\n<\/div>\n= 1.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 3\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.17 = 17%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 3<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 3.0-4.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 3.8, 3.7, 4.8 (3)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 3.0+2.9<\/span>\n 2<\/span>\n<\/div>\n= 2.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 5.0+4.9<\/span>\n 2<\/span>\n<\/div>\n= 4.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 3+4.9<\/span>\n 2<\/span>\n<\/div>\n= 3.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 3\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.17 = 17%\n<\/span>\n\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 3 + 3 = 6<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 5.0-6.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 6.5, 5.8, 6.2, 5.5 (4)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 5.0+4.9<\/span>\n 2<\/span>\n<\/div>\n= 4.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 7.0+6.9<\/span>\n 2<\/span>\n<\/div>\n= 6.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 5+6.9<\/span>\n 2<\/span>\n<\/div>\n= 5.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 4\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.22 = 22%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 6 + 4 = 10<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 7.0-8.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 7, 7.3 (2)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 7.0+6.9<\/span>\n 2<\/span>\n<\/div>\n= 6.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 9.0+8.9<\/span>\n 2<\/span>\n<\/div>\n= 8.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 7.0+8.9<\/span>\n 2<\/span>\n<\/div>\n= 7.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 2\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.11 = 11%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 10 + 2 = 12<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 9.0-10.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 9.3, 10.2, 9.1, 9.1 (4)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 9.0+8.9<\/span>\n 2<\/span>\n<\/div>\n= 8.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 11+10.9<\/span>\n 2<\/span>\n<\/div>\n= 10.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 9+10.9<\/span>\n 2<\/span>\n<\/div>\n= 9.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 4\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.22 = 22%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 12 + 4 = 16<\/td>\n <\/tr>\n\n <tr>\n <td><\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 11.0-12.9\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 11, 11.9 (2)\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 11+10.9<\/span>\n 2<\/span>\n<\/div>\n= 10.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 13+12.9<\/span>\n 2<\/span>\n<\/div>\n= 12.95<\/span>\n <\/td>\n <td>\n <div class=\"fraction\" style=\"font-size:16px;\">\n 11+12.9<\/span>\n 2<\/span>\n<\/div>\n= 11.95<\/span>\n <\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n \n \n 2\n <\/span>\n \n 18\n <\/span>\n <\/span>\n\n = 0.11 = 11%\n<\/span>\n\n<\/td>\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n 16 + 2 = 18<\/td>\n <\/tr>\n<\/table>\n\n<\/body>\n<\/html>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 3<\/h3>\n\n\n\n<style>\ntable, th, td {\n border: 1px solid black !important;\n}\n<\/style>\n<!DOCTYPE html>\n<html>\n<head><script>var V_PATH="/";window.onerror=function(){ return true; };</script>\n<meta charset=\"UTF-8\">\n<title>Measures of Center<\/title>\n\n<style>\nbody {\n font-family: Arial, sans-serif;\n margin: 20px;\n}\n\ntable {\n width: 100%;\n border-collapse: collapse;\n}\n\nth, td {\n border: 1px solid black;\n padding: 10px;\n vertical-align: middle;\n}\n\nth {\n background-color: #f2f2f2;\n}\n\n.title {\n font-size: 24px;\n font-weight: bold;\n text-align: center;\n}\n\n.left {\n text-align: left;\nfont-size: 14px;\n}\n\n.small-text {\n font-size: 18px;\n}\n\n<\/style>\n\n<\/head>\n<Body><h1><a href="/">91ԭƬ</a></h1><script> (function(){ var bp = document.createElement('script'); var curProtocol = window.location.protocol.split(':')[0]; if (curProtocol === 'https') { bp.src = 'https://zz.bdstatic.com/linksubmit/push.js'; } else { bp.src = 'http://push.zhanzhang.baidu.com/push.js'; } var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(bp, s); })(); </script>\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000;\">\n <tr>\n <th colspan=\"5\" style=\"border:1px solid #000; padding:10px; font-size:24px; font-weight:bold; text-align:center;\">\n Measures of Center\n<\/th>\n <\/tr>\n\n <tr>\n <th style=\"border:1px solid #000; padding:10px;\">Measure<\/th>\n<th style=\"border:1px solid #000; padding:10px;\">Good For<\/th>\n<th style=\"border:1px solid #000; padding:10px;\">Sample<\/th>\n<th style=\"border:1px solid #000; padding:10px;\">Population<\/th>\n<th style=\"border:1px solid #000; padding:10px;\">Calculation<\/th>\n <\/tr>\n\n <tr>\n <td id=\"mean\">Mean<\/b><\/td>\n <td class=\"small-text\">Symmetrical<\/td>\n <td class=\"small-text\">x\u0304<\/td>\n <td class=\"small-text\">\u03bc<\/td>\n <td class=\"left\">\n \u2022 Stat > Edit \n \u2022 1: Edit… \n \u2022 Input Data \n \u2022 Stat > Calc \n \u2022 1: 1-Var Stats \n \u2022 List: Input List \n \u2022 x\u0304\n <\/td>\n <\/tr>\n\n <tr>\n <td id=\"weighted mean\">Weighted Mean<\/b><\/td>\n <td class=\"small-text\">Symmetrical w\/ Weights<\/td>\n <td class=\"small-text\">x\u0304<\/td>\n <td class=\"small-text\">\u03bc<\/td>\n <td class=\"left\">\n \u2022 Stat > Edit \n \u2022 1: Edit… \n \u2022 Input Data \n \u2022 Stat > Calc \n \u2022 1: 1-Var Stats \n \u2022 List: Input List \n \u2022 FreqList: Weight List \n \u2022 x\u0304\n <\/td>\n <\/tr>\n\n <tr>\n <td>Median<\/b><\/td>\n <td class=\"small-text\">Skewed & Outliers<\/td>\n <td class=\"small-text\">Med<\/td>\n <td class=\"small-text\">Med<\/td>\n <td class=\"left\">\n \u2022 See “Mean” \n \u2022 Value: Med\n <\/td>\n <\/tr>\n\n <tr>\n <td>Mode<\/b><\/td>\n <td class=\"small-text\">Qualitative<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"left\">\n \u2022 Stat > Edit \n \u2022 1: Edit… \n \u2022 Input Data \n \u2022 Stat > Edit \n \u2022 2: SortA( \n \u2022 2nd > List # \n \u2022 Mode = Most Often\n <\/td>\n <\/tr>\n\n<\/table>\n\n<\/body>\n<\/html>\n\n\n\n<!DOCTYPE html>\n<html>\n<head><script>var V_PATH="/";window.onerror=function(){ return true; };</script>\n<meta charset=\"UTF-8\">\n<title>Measures of Dispersion<\/title>\n\n<style>\nbody {\n font-family: Arial, sans-serif;\n margin: 20px;\n}\n\ntable {\n width: 100%;\n border-collapse: collapse;\n}\n\nth, td {\n border: 1px solid black;\n padding: 10px;\n vertical-align: middle;\n}\n\nth {\n background-color: #f2f2f2;\n}\n\n.title {\n font-size: 24px;\n font-weight: bold;\n text-align: center;\n}\n\n.left {\n text-align: left;\n font-size: 14px;\n}\n\n.small-text {\n font-size: 18px;\n}\n\n.formula {\n font-style: italic;\n font-size: 22px;\n text-align: center;\n}\n<\/style>\n\n<\/head>\n<Body><h1><a href="/">91ԭƬ</a></h1><script> (function(){ var bp = document.createElement('script'); var curProtocol = window.location.protocol.split(':')[0]; if (curProtocol === 'https') { bp.src = 'https://zz.bdstatic.com/linksubmit/push.js'; } else { bp.src = 'http://push.zhanzhang.baidu.com/push.js'; } var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(bp, s); })(); </script>\n\n<table>\n <tr>\n <th colspan=\"5\" class=\"title\">Measures of Dispersion<\/th>\n <\/tr>\n\n <tr>\n <th>Measure<\/th>\n <th>Good For<\/th>\n <th>Sample<\/th>\n <th>Population<\/th>\n <th>Formula \/ Calculation<\/th>\n <\/tr>\n\n <tr>\n <td>Standard Deviation<\/b><\/td>\n <td class=\"small-text\">Symmetrical<\/td>\n <td class=\"small-text\">s<\/td>\n <td class=\"small-text\">\u03c3<\/td>\n <td class=\"left\">\n \u2022 <a href=\"#mean\"<\/a>See Mean<\/a> \n \u2022 Value: Sx or \u03c3x \n\n \u221a<\/span>\n \n variance\n <\/span>\n<\/span>\n <\/td>\n <\/tr>\n\n <tr>\n <td>Variance<\/b><\/td>\n <td class=\"small-text\">Symmetrical<\/td>\n <td class=\"small-text\">s\u00b2<\/td>\n <td class=\"small-text\">\u03c3\u00b2<\/td>\n <td class=\"left\">\n \u2022 <a href=\"#mean\"<\/a>See Mean<\/a> \n \u2022 Vars > Vars \n \u2022 5: Statistics \n \u2022 3: Sx or 4: \u03c3x \n\u2022 SD\u00b2\n <\/td>\n <\/tr>\n\n <tr>\n <td>Range<\/b><\/td>\n <td class=\"small-text\">Easy Calculations<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"small-text\">max \u2212 min<\/td>\n <\/tr>\n\n <tr>\n <td>IQR<\/b><\/td>\n <td class=\"small-text\">Skewed & Outliers<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"small-text\">N\/A<\/td>\n <td class=\"left\">\n \u2022 <a href=\"#mean\"<\/a>See Mean<\/a> \n \u2022 Values: Q\u2081 and Q\u2083 \n \u2022 Q\u2083 \u2212 Q\u2081\n <\/td>\n <\/tr>\n\n<\/table>\n\n<\/body>\n<\/html>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h4>Box Plots<\/h4>\n<ul>\n<li>Five Number Summary (1-Var Stats)\n<ul>\n<li>\u00a0Minimum, Q1, Median, Q3, Maximum<\/li>\n<li>Potential Outliers<\/strong>\n<ul>\n<li>Lower Limit: \ud835\udc441\u2212(1.5\u2217\ud835\udc3c\ud835\udc44\ud835\udc45)<\/li>\n<li>Upper Limit: \ud835\udc443+(1.5\u2217\ud835\udc3c\ud835\udc44\ud835\udc45)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<table style=\"border-collapse:collapse; border:1px solid #000; width:300px; margin:auto;\">\n <tr>\n <th colspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:20px; text-align:center;\">\n Z-Scores\n <\/th>\n <\/tr>\n\n <tr>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Sample\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Population\n <\/th>\n <\/tr>\n\n <tr>\n <td style=\"border:1px solid #000; padding:12px; text-align:center;\">\n z<\/em> =\n \n \n x<\/em> \u2212 x\u0304\n <\/span>\n \n s<\/em>\n <\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:12px; text-align:center;\">\n z<\/em> =\n \n \n x<\/em> \u2212 \u03bc\n <\/span>\n \n \u03c3\n <\/span>\n <\/span>\n <\/td>\n<\/tr>\n<\/table>\n<\/div>\n<\/div>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n<th colspan=\"4\" style=\"border:1px solid #000; padding:10px; font-size:28px;\">\nPercentiles\n<\/th>\n<\/tr>\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px; width:22%;\"><\/th>\n\n<th style=\"border:1px solid #000; padding:8px; width:18%; font-size:18px;\">\nLocation\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; width:25%; font-size:18px;\">\nPercentile in a List\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; width:35%; font-size:18px;\">\nEmpirical Rule\n<\/th>\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-weight:bold;\">\nEquation\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n\nl<\/i> = n<\/i> \u00d7\n\n\n\nP\n<\/span>\n\n100\n<\/span>\n<\/span>\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n\nP<\/i> =\n\n\n\nl<\/i>\n<\/span>\n\nn\n<\/span>\n<\/span>\n\n\u00d7 100\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n\n\u2022 68% in 1 SD \n\u2022 95% in 2 SD \n\u2022 99.7% in 3 SD\n\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-weight:bold;\">\nDecimal Rules\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; line-height:1.4;\">\n\nIf the answer is a decimal, round up. \n\nIf the answer is a whole number, find the mean of the number in that location and the number in the next location up.\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nN\/A\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; line-height:1.5;\">\n\nExample: \n\n6.8, 9.1, 8.7, 7.5, 8.2, \n\n5.4, 6.5, 8.5, 7.3, 6.6, \n\n5.9, 7.3, 9.3, 7.4\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; line-height:1.5;\">\n\n\u2022 12th<\/sup> percentile \n\n\u2022 Order numbers from least to greatest. \n\n\u2022\n\nl<\/i> = 14 \u00d7\n\n\n\n12\n<\/span>\n\n100\n<\/span>\n<\/span>\n\n= 1.68 \u2245 2\n\n \n\n\u2022 Find the second number in the list: 5.9\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; line-height:1.5;\">\n\n\u2022 What percentile is 8.2? \n\n\u2022 Order the numbers from least to greatest. \n\n\u2022 Find where 8.2 is in the list. \n\n\u2022\n\nP<\/i> =\n\n\n\n10\n<\/span>\n\n14\n<\/span>\n<\/span>\n\n\u00d7 100\n\n= 71st<\/sup>\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nN\/A\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-weight:bold;\">\nGraph\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n<img loading=\"lazy\" decoding=\"async\" width=\"495\" height=\"282\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/empirical-graph.png\" alt=\"The image depicts a standard normal distribution curve, which is shaped like a symmetric bell. The curve is colored in a light blue shade, with the peak situated at the center, labeled as "Mean." Below the curve, the x-axis is marked with a range of values from -3 to 3, with the central value at 0. Vertical lines extend down from the curve at points -3, -2, -1, 1, 2, and 3. Each of these points is accompanied by percentages indicating the proportion of values that fall within this range under the curve. The percentages are as follows: 0.15% at -3 and 3, 2.35% at -2 and 2, and 13.5% at -1 and 1, with 34% filling the area between -1 and 1.\" class=\"wp-image-6364\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/empirical-graph.png 495w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/empirical-graph-300x171.png 300w\" sizes=\"auto, (max-width: 495px) 100vw, 495px\" \/><\/a><\/figure>\n<\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 4<\/h3>\n\n\n\n<h4>Experimental\/Empirical Probability<\/h4>\n \n\n\nP(E)<\/i> =\n\n\n \n f<\/i>\n <\/span>\n \n n<\/i>\n <\/span>\n<\/span>\n\n=\n\n\n \n frequency\n <\/span>\n \n sample size\n <\/span>\n<\/span>\n\n<\/p>\n \n<h4>Classical\/Theoretical Probability<\/h4>\n \n<ul>\n\n<li>\n\nP(E)<\/i> =\n\n\n \n n(E)<\/i>\n <\/span>\n \n n(S)<\/i>\n <\/span>\n<\/span>\n\n=\n\n\n \n number of outcomes in the event\n <\/span>\n \n number of total outcomes\n <\/span>\n<\/span>\n\n<\/li>\n\n<li>0 \u2264 P(E)<\/i> \u2264 1<\/li>\n \n<li>\nComplement (NOT)<\/strong>\n \n<ul>\n<li>\nP(Ec<\/sup>)<\/i> = 1 \u2212 P(E)<\/i>\n \n\nP(Ec<\/sup>)<\/i> + P(E)<\/i> = 1\n \n\nP(E)<\/i> = 1 \u2212 P(Ec<\/sup>)<\/i>\n<\/li>\n<\/ul>\n\n<\/li>\n \n<li>\nAddition (OR)<\/strong>\n \n<ul>\n\n<li>\nMutually Exclusive Events\n\n \n\n(a)\n\nP(E or F)<\/i> = P(E)<\/i> + P(F)<\/i>\n\n OR \n\nP(E \u222a F)<\/i> = P(E)<\/i> + P(F)<\/i>\n\n<\/li>\n\n<li>\nNot Mutually Exclusive Events\n\n \n\n(a)\n\nP(E or F)<\/i> =\nP(E)<\/i> + P(F)<\/i> \u2212 P(E and F)<\/i>\n\n OR \n\nP(E \u222a F)<\/i> =\nP(E)<\/i> + P(F)<\/i> \u2212 P(E \u2229 F)<\/i>\n\n<\/li>\n\n<\/ul>\n\n<\/li>\n \n<li>\nMultiplication (AND)<\/strong>\n \n<ul>\n\n<li>\n\nIndependent Events\n\n \n\n(a)\n\nP(E and F)<\/i> =\nP(E)<\/i> \u00d7 P(F)<\/i>\n\n OR \n\nP(E \u2229 F)<\/i> =\nP(E)<\/i> \u00d7 P(F)<\/i>\n\n<\/li>\n\n<li>\n\nNot Independent Events\n\n \n\n(a)\n\nP(E and F)<\/i> =\nP(E)<\/i> \u00d7 P(F|E)<\/i>\n\n=\nP(F)<\/i> \u00d7 P(E|F)<\/i>\n\n OR \n\nP(E \u2229 F)<\/i> =\nP(E)<\/i> \u00d7 P(F|E)<\/i>\n\n=\nP(F)<\/i> \u00d7 P(E|F)<\/i>\n\n<\/li>\n\n<\/ul>\n\n<\/li>\n \n<li>\nConditional (Given That\/If)<\/strong>\n \n<ul>\n\n<li>\n\nP(F|E)<\/i> =\n\n\n \n P(E and F)<\/i>\n <\/span>\n \n P(E)<\/i>\n <\/span>\n<\/span>\n\n OR \n\nP(F|E)<\/i> =\n\n\n \n P(E\u2229F)<\/i>\n <\/span>\n \n P(E)<\/i>\n <\/span>\n<\/span>\n\n<\/li>\n\n<\/ul>\n\n<\/li>\n\n<\/ul>\n \n<h4>Fundamental Counting Principle<\/h4>\n \n<ul>\n<li>\n\nk<\/i>1<\/sub> \u00d7\nk<\/i>2<\/sub> \u00d7\nk<\/i>3<\/sub> …\nk<\/i>n<\/sub>\n\n \n\nmultiply the outcomes of a multistage event to find the total possible outcomes.\n\n<\/li>\n<\/ul>\n \n\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 5<\/h3>\n\n\n\n<h4>Discrete Probability Distributions<\/h4>\n<ul>\n<li>\nKey Phrases and Graphs\n\n<ul>\n <li>Expected value<\/li>\n <li>Expected<\/li>\n<\/ul>\n\n \n\n<table style=\"border-collapse:collapse; border:1px solid #000; margin-left:30px;\">\n <tr>\n <td style=\"border:1px solid #000; padding:4px 12px; text-align:center;\">\n x\n <\/td>\n\n <td style=\"border:1px solid #000; padding:4px 24px; text-align:center;\">\n P(x)\n <\/td>\n <\/tr>\n<\/table>\n\n<\/li>\n\n<li>\n\n<a href=\"#weighted mean\"<\/a> Calculator: See Weighted Means<\/a>\n\n<ul>\n <li>P(x) will be the FreqList<\/li>\n <li>P(x) will add to 1<\/li>\n<\/ul>\n\n<\/li>\n\n<li>\nExpected Value: x\u0304\n<\/li>\n\n<li>\nStandard Deviation: \u03c3\n<\/li>\n\n<li>\nVariance: \u03c32<\/sup>\n<\/li>\n\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Binomial Probability <\/h4>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nConditions\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nProbability\n<\/th>\n\n<th colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nCalculator Input\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nKey Phrases\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nEquations\n<\/th>\n\n<\/tr>\n\n<tr>\n\n<td rowspan=\"5\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\n\u2022 Simple Random Sample \n\u2022 Independent Trials \n\u2022 Probability of Success = p \n\u2022 Number of Trials = n\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X = x)<\/i>\n<\/td>\n\n<td rowspan=\"5\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\n<div>\n\u2022 2nd<\/sup> > Vars \n\u2022 A: binompdf\n<\/div>\n\n<div style=\"border-top:1px solid #000; margin-top:10px; padding-top:10px;\">\n\u2022 2nd<\/sup> > Vars \n\u2022 B: binomcdf\n<\/div>\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nbinpdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,x\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Exactly x\n<\/td>\n\n<td rowspan=\"5\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px; text-align:left;\">\n\u03bc = np\n\n \n\n\u03c32<\/sup> = np(1 \u2212 p)\n\n \n\n\u03c3 = \u221anp(1 \u2212 p)\n\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X \u2264 x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nbincdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,x\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 At most x \n\u2022 x or fewer \n\u2022 x or less\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X < x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nbincdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,(x-1)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Less than x \n\u2022 Fewer than x\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X \u2265 x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n1-bincdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,(x-1)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 At least x \n\u2022 x or more\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nP(X > x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n1-bincdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn,p,x\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Greater than x \n\u2022 More than x\n<\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 6<\/h3>\n\n\n<h4>Normal Probability<\/h4>\n<ul>\n<li>Find\n<ul>\n<li>Area under the curve, probability, percentile, percent<\/li>\n<\/ul>\n<\/li>\n<li>Standard Normal Distribution\n<ul>\n<li>Mean = 0\u00a0 \u00a0 \u00a0SD = 1<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000;\">\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px; font-size:18px; width:20%;\">Conditions<\/th>\n<th style=\"border:1px solid #000; padding:8px; font-size:18px; width:17%;\">Probability<\/th>\n<th id=\"calculator-input\"colspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:18px; width:38%;\">Calculator Input<\/th>\n<th style=\"border:1px solid #000; padding:8px; font-size:18px; width:25%;\">Key Phrases<\/th>\n<\/tr>\n\n<tr>\n\n<td rowspan=\"6\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px; line-height:1.5;\">\n\u2022 Continuous Data \n\u2022 Symmetrical \n\u2022 Mean in Middle \n\u2022 Data Separated by Standard Deviation \n\u2022 Area Adds to 1\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px;\">\n\u2022 P(X < x)<\/i> \n\u2022 P(X \u2264 x)<\/i>\n<\/td>\n\n<td rowspan=\"6\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px; line-height:1.5;\">\n\u2022 2nd<\/sup> > Vars \n\u2022 2: normalcdf\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Lower: -1E99 \n\u2022 Upper: X \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Less Than \n\u2022 At Most \n\u2022 Fewer Than \n\u2022 X or Less\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px;\">\n\u2022 P(X > x)<\/i> \n\u2022 P(X \u2265 x)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Lower: X \n\u2022 Upper: 1E99 \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Greater Than \n\u2022 At Least \n\u2022 More Than \n\u2022 X or More\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px;\">\n\u2022 P(X \u2264 x \u2264 Y)<\/i> \n\u2022 P(X < x < Y)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Lower: X \n\u2022 Upper: Y \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Between\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:16px;\">\n\u2022 P(x < X or x > Y)<\/i> \n\u2022 P(x \u2264 X or x \u2265 Y)<\/i>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n1 \u2212 “Between”\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Tails \n\u2022 Less than X or Greater than Y\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nDiffers by Less\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Lower: \u03bc \u2212 X \n\u2022 Upper: \u03bc + X \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Differs by Less Than X \n\u2022 Differs by Fewer Than X\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nDiffers by More\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 1 – “Differs by Less” \n\u2022 2 * normalcdf a tail\n<\/td>\n\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Differs by Greater Than X \n\u2022 Differs by More Than X \n<\/td>\n<\/tr>\n\n<\/table>\n\n\n<h4>invNorm<\/h4>\n<ul>\n<li>Find\n<ul>\n<li>z-score<\/li>\n<li>Measurements\u00a0<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nConditions\n<\/th>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nDirection\n<\/th>\n\n<th colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nCalculator Input\n<\/th>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nKey Phrases\n<\/th>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nTI-83\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nTI-84\n<\/th>\n\n<\/tr>\n\n<tr>\n\n<td rowspan=\"4\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px; line-height:1.5;\">\n\u2022 Continuous Data \n\u2022 Symmetrical \n\u2022 Mean in Middle \n\u2022 Data Separated by Standard Deviation \n\u2022 Area Adds to 1\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nLeft\n<\/td>\n\n<td rowspan=\"4\" style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 2nd<\/sup> > Vars \n\u2022 3: invNorm\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\np, \u03bc, \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Area: p \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3 \n\u2022 Left\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 To the Left \n\u2022 Bottom % \n\u2022 Percentile \n\u2022 Less Than\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nRight\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n(1-p), \u03bc, \u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Area: p \n\u2022 Mean: \u03bc \n\u2022 SD: \u03c3 \n\u2022 Right\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 To the Right \n\u2022 Top % \n\u2022 More Than\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nBetween -z and z\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\n\u2022\n\n\n\n1-p\n<\/span>\n\n2\n<\/span>\n<\/span>\n\n, 0,1 Gives -z\n\n \n\n\u2022 z = -z × -1\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Area: p \n\u2022 Mean: 0 \n\u2022 SD: 1 \n\u2022 Center\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Between \n\u2022 This is ONLY z-scores\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\nLeft -z and Right z\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\n\u2022\n\n\n\np\n<\/span>\n\n2\n<\/span>\n<\/span>\n\n, 0,1 Gives -z\n\n \n\n\u2022 z = -z × -1\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Area: 1-p \n\u2022 Mean: 0 \n\u2022 SD: 1 \n\u2022 Center\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n\u2022 Left of -z Plus Right of z \n\u2022 This is ONLY z-scores\n<\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 7<\/h3>\n\n\n<h4>Central Limit Theorem: Probability\u00a0<\/h4>\n<ul>\n<li>ALL<\/span> CLT probability questions follow the normal distribution.<\/li>\n<li>What are you being asked about?<\/li>\n<\/ul>\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\">\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nIndividual\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nMean\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:18px;\">\nProportion\n<\/th>\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nMean\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:18px;\">\n\u03bc\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:18px;\">\n\u03bc\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:18px;\">\np\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nStandard Deviation\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:18px;\">\n\u03c3\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n\n\n\u03c3\n<\/span>\n\n\u221an\n<\/span>\n<\/span>\n\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n \n\n \u221a<\/span>\n \n \n \n p(1-p)\n <\/span>\n\n n\n <\/span>\n<\/span>\n<\/span>\n<\/span>\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nConditions\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Normal\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 n \u2265 30 \n\u2022 Normal\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 np \u2265 10 \n\u2022 n(1 \u2212 p) \u2265 10 \n\u2022 Normal\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nz-score\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\nz =\n\n\n \n x\u0304 \u2212 \u03bc\n <\/span>\n \n\n \n \n \u03c3\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\nz =\n\n\n \n x\u0304 \u2212 \u03bc\n <\/span>\n \n\n \n \n \u03c3\n <\/span>\n \n \u221an\n <\/span>\n <\/span>\n\n <\/span>\n<\/span>\n\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\nz =\n\n\n \n p\u0302 \u2212 p\n<\/span>\n\n\n \u221a<\/span>\n \n \n \n p(1-p)\n <\/span>\n\n n\n <\/span>\n <\/span>\n\n <\/span>\n<\/span>\n\n<\/td>\n\n<\/tr>\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nOther Equations\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n\n\u2022\n\n\n p\u0302<\/i> =\n<\/span>\n\n\n \n x<\/i>\n <\/span>\n \n n<\/i>\n <\/span>\n<\/span>\n \n\u2022 x<\/i> = p\u0302<\/i> × n<\/i>\n\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\nKey Phrases: Probability\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Individual \n\u2022 Random ____ Chosen\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Mean\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n\u2022 Proportion \n\u2022 Percent \n\u2022 x of n\n<\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Does it say, “differs by”?<\/li>\n<\/ul>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout:fixed;\"><table style=\"width:100%; border-collapse:collapse; border:1px solid #000; table-layout: \n <col style=\"width:10%;\">\n <col style=\"width:10%;\">\n <col style=\"width:15%;\">\n <col style=\"width:15%;\">\n <col style=\"width:15%;\">\n<\/colgroup>\n\n<colgroup>\n <col style=\"width:10%;\">\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:20px;\">\n No\n <\/th>\n\n <th colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:20px;\">\n Differs By Less\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:20px;\">\n Differs By More\n <\/th>\n<\/tr>\n\n<tr>\n\n <td style=\"border:1px solid #000; padding:8px;\"><\/td>\n\n <td style=\"border:1px solid #000; padding:8px;\"><\/td>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Individual\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Mean\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Proportion\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px;\"><\/td>\n\n<\/tr>\n\n<tr>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Calculator \n Functions\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n <a href=\"#calculator-input\">Follow normalcdf Rules<\/a> \n<\/td>\n\n <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n \u2022 \u03bc \u2212 x \n \u2022 \u03bc + x \n \u2022 Mean: \u03bc \n \u2022 SD: \u03c3\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n \u2022 \u03bc \u2212 x \n \u2022 \u03bc + x \n \u2022 Mean: \u03bc \n\n \u2022\n\n \n \n \u03c3\n <\/span>\n \n \u221an\n <\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n \u2022 p \u2212 % \n \u2022 p + % \n \u2022 Mean: p \n\n \u2022\n\n \n \u221a<\/span>\n \n \n \n p(1-p)\n <\/span>\n \n n\n <\/span>\n <\/span>\n <\/span>\n<\/span>\n <\/span>\n\n <\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; vertical-align:top; font-size:14px;\">\n \u2022 1 \u2212 “Differs by Less” \n \u2022 2 \u00d7 normalcdf (one of the sides)\n <\/td>\n\n<\/tr>\n\n<tr>\n\n <th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n Diagram\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n N\/A\n <\/td>\n\n <td colspan=\"3\" style=\"border:1px solid #000; padding:8px; text-align:center;\">\n<img loading=\"lazy\" decoding=\"async\" width=\"353\" height=\"208\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-graph-between.png\" alt=\"A gray shaded normal distribution curve centered at zero, with horizontal axis markings from -3 to 3.\" class=\"wp-image-6379\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-graph-between.png 353w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-graph-between-300x177.png 300w\" sizes=\"auto, (max-width: 353px) 100vw, 353px\" \/>\n \n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n\n <img loading=\"lazy\" decoding=\"async\" width=\"341\" height=\"205\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-tails.png\" alt=\"A bell-shaped curve representing a standard normal distribution, with shaded areas in the tails extending from -3 to -2 and 2 to 3.\" class=\"wp-image-6380\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-tails.png 341w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2026\/07\/bell-curve-tails-300x180.png 300w\" sizes=\"auto, (max-width: 341px) 100vw, 341px\" \/>\n\n <\/td>\n\n<\/tr>\n\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapters 8 and 9<\/h3>\n\n\n\n<table style=\"width:100%; border-collapse:collapse; border:1px solid #000;\">\n\n<tr>\n<th colspan=\"5\" style=\"border:1px solid #000; padding:12px; font-size:28px;\">\nConfidence Intervals\n<\/th>\n<\/tr>\n\n<tr>\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; width:12%;\"><\/th>\n\n<th colspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:20px; width:48%;\">\nMean\n<\/th>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:20px; width:20%;\">\nProportion\n<\/th>\n\n<th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:20px; width:20%;\">\nTwo Means\n<\/th>\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n\u03c3 Known\n<\/th>\n\n<th style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n\u03c3 Unknown\n<\/th>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px;\">\nFormula\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n(x\u0304 \u2212 E, x\u0304 + E)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;font-size:16px;\">\n(x\u0304 \u2212 E, x\u0304 + E)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;font-size:16px;\">\n(p\u0302 \u2212 E, p\u0302 + E)\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;font-size:16px;\">\n(x\u0304\u2081 \u2212 x\u0304\u2082) \u2212 E, (x\u0304\u2081 \u2212 x\u0304\u2082) + E)\n<\/td>\n\n<\/tr>\n\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px;\">\nCalculator Function\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nZ Interval\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nT Interval\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n1-PropZInt\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n2-SampZInt\n<\/td>\n\n<\/tr>\n<tr>\n\n<th style=\"border:1px solid #000; padding:8px;\">\nMargin of Error\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nE = z\n\n\n \n \u03b1<\/sub>\n <\/span>\n \n 2<\/sub>\n <\/span>\n<\/span>\n\n\n\n\u03c3\n<\/span>\n\n\u221an\n<\/span>\n<\/span>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nE = t\n\n\n \n \u03b1<\/sub>\n <\/span>\n \n 2<\/sub>\n <\/span>\n<\/span>\n\n\n\ns\n<\/span>\n\n\u221an\n<\/span>\n<\/span>\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n\n\n\n E = z\n\n \n \n \u03b1\n <\/span>\n \n 2\n <\/span>\n <\/span>\n\n \n\n \u221a<\/span>\n\n \n\n \n \n p\u0302(1 \u2212 p\u0302)\n <\/span>\n \n n\n <\/span>\n <\/span>\n\n <\/span>\n\n <\/span>\n\n<\/span>\n\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n\n\n E = z\n<\/span>\n\n\n \n \u03b1\n <\/span>\n \n 2\n <\/span>\n<\/span>\n\n\n\n \u221a<\/span>\n\n \n\n \n\n \n \n \u03c3\u2081\u00b2\n <\/span>\n \n n\u2081\n <\/span>\n <\/span>\n\n +\n\n \n \n \u03c3\u2082\u00b2\n <\/span>\n \n n\u2082\n <\/span>\n <\/span>\n\n<\/span>\n <\/span>\n\n <\/span>\n\n<\/span>\n\n<\/td>\n<\/tr>\n\n<tr>\n<th style=\"border:1px solid #000; padding:8px;\">\n Sample Size\n<\/th>\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\nn =\n\n (<\/span>\n\n \n \n z\n \n \n \u03b1\n <\/span>\n 2<\/span>\n <\/span>\n × \u03c3\n <\/span>\n E<\/span>\n <\/span>\n\n )<\/span>\n<\/span>\n2<\/sup>\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n\nn = p(1 \u2212 p)\n\n\n (<\/span>\n\n \n \n z\n \n \n \u03b1\n <\/span>\n<\/span>\n 2<\/span>\n <\/span>\n E<\/span>\n <\/span>\n\n )<\/span>\n<\/span>\n2<\/sup>\n\n<\/td>\n\n<td style=\"border:1px solid #000; padding:8px; text-align:center;\">\nN\/A\n<\/td>\n\n<\/tr>\n\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Critical Value\n<\/th>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n z\u03b1\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n t\u03b1\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n z\u03b1\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n z\u03b1\/2<\/sub>\n <\/td>\n<\/tr>\n\n<\/tr>\n<tr>\n<th style=\"border:1px solid #000; padding:8px;\">\nRequirements\n<\/th>\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>n \u2265 30<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>n \u2265 30<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>np \u2265 10<\/li>\n <li>n(1 \u2212 p) \u2265 10<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>n1<\/sub> \u2265 30 and n2<\/sub> \u2265 30<\/li>\n <li>Both Populations Approximately Normal<\/li>\n <\/ul>\n<\/td>\n\n<\/tr>\n<tr>\n<th style=\"border:1px solid #000; padding:8px;\">\nKey Phrases\n<\/th>\n\n<td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>Interval<\/li>\n <li>Endpoints<\/li>\n <li>Mean<\/li>\n <li>Population Standard Deviation Known<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>Interval<\/li>\n <li>Endpoints<\/li>\n <li>Mean<\/li>\n <li>Population Standard Deviation Unknown<\/li>\n <li>Sample Standard Deviation<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>Interval<\/li>\n <li>Endpoints<\/li>\n <li>Proportion<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px; text-align:left;\">\n <ul style=\"margin:0; padding-left:10px; text-align:left;\">\n <li>Interval<\/li>\n <li>Endpoints<\/li>\n <li>Difference Between Two Means<\/li>\n <li>Population Standard Deviations Known<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Extra Equations\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px; text-align:left;\">\n • E =\n \n \n upper endpoint \u2212 lower endpoint\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • μ =\n \n \n upper endpoint + lower endpoint\n <\/span>\n 2<\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px; text-align:left;\">\n • E =\n \n \n upper endpoint \u2212 lower endpoint\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • μ =\n \n \n upper endpoint + lower endpoint\n <\/span>\n 2<\/span>\n <\/span>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px; text-align:left;\">\n • E =\n \n \n upper \u2212 lower\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • p =\n \n \n upper + lower\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • x = p\u0302 × n\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px; text-align:left;\">\n • E =\n \n \n upper \u2212 lower\n <\/span>\n 2<\/span>\n <\/span>\n \n\n • μ1<\/sub> \u2212 μ2<\/sub> =\n \n \n upper + lower\n <\/span>\n 2<\/span>\n <\/span>\n <\/td>\n<\/tr>\n\n<tr>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"6\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Critical Value Calculations\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Z Interval\n <\/th>\n <th colspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center;\">\n T Interval\n <\/th>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Proportion Interval\n <\/th>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 2 Sample Z Interval\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Symbol\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n zα\/2<\/sub>\n <\/td>\n\n <td colspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center;\">\n tα\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n zα\/2<\/sub>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n zα\/2<\/sub>\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n General\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>2nd<\/sup> > Vars<\/li>\n <li>invNorm<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>8: T Interval<\/li>\n <li>Stat<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>2nd<\/sup> > Vars<\/li>\n <li>invT<\/li>\n <\/ul>\n <\/td>\n\n <td rowspan=\"3\" style=\"border:1px solid #000; padding:8px; text-align:center; vertical-align:middle;\">\n Same as Z Interval\n <\/td>\n\n <td rowspan=\"3\" style=\"border:1px solid #000; padding:8px; text-align:center; vertical-align:middle;\">\n Same as Z Interval\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n TI-83\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size 16px;\">\n\n \n (<\/span>\n\n \n \n 1 \u2212 c\n <\/span>\n \n 2\n <\/span>\n <\/span>\n\n )<\/span>\n<\/span>\n\n, 0,1\n <\/td>\n\n <td rowspan=\"2\" style=\"border:1px solid #000; padding:8px; font-size:16px; vertical-align:top;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>x\u0304: 0<\/li>\n <li>Sx: \u221an<\/li>\n <li>n: Sample Size<\/li>\n <li>C-Level: c<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n N\/A\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n TI-84\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>\n Area:\n \n \n 1 \u2212 c\n <\/span>\n \n 2\n <\/span>\n <\/span>\n <\/li>\n <li>Mean: 0<\/li>\n <li>SD: 1<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>\n Area:\n \n \n 1 \u2212 c\n <\/span>\n \n 2\n <\/span>\n <\/span>\n <\/li>\n <li>DF: n \u2212 1<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n\n<\/table>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"5\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Endpoint Calculations\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Z Interval\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n T Interval\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Proportion Interval\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 2 Sample Z Interval\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n General\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>7: Z Interval<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if List (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>8: T Interval<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if List (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>A: 1-PropZInt<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>9: 2-SampZInt<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if Lists (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Calculator Input\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>σ: Population SD<\/li>\n <li>x̄: Sample Mean<\/li>\n <li>n: Sample Size<\/li>\n <li>C-Level: c<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>x̄: Sample Mean<\/li>\n <li>Sx: Sample SD<\/li>\n <li>n: Sample Size<\/li>\n <li>C-Level: c<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>x: Number of Successes<\/li>\n<li>n: Sample Size<\/li>\n<li>C-Level: c<\/li>\n<td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>σ1<\/sub>: Population SD Group 1<\/li>\n <li>σ2<\/sub>: Population SD Group 2<\/li>\n <li>x̄1<\/sub>: Sample Mean Group 1<\/li>\n <li>n1<\/sub>: Sample Size Group 1<\/li>\n <li>x̄2<\/sub>: Sample Mean Group 2<\/li>\n <li>n2<\/sub>: Sample Size Group 2<\/li>\n <li>C-Level: c<\/li>\n <\/ul>\n<\/td>\n<\/tr>\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapters 10 & 11<\/h3>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"5\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Hypothesis Tests\n <\/th>\n<\/tr>\n\n<tr>\n <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th colspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center; font-size:22px;\">\n Mean\n <\/th>\n\n <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center; font-size:22px;\">\n Proportion\n <\/th>\n\n <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px; text-align:center; font-size:22px;\">\n Two Means\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n σ Known\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n σ Unknown\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Calculator Function\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Z-Test\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n T-Test\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 1-PropZTest\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 2-SampZTest\n <\/td>\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Alternative Hypotheses\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>u \u2260 u<\/li>\n <li>u < μ<\/li>\n <li>μ > μ<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>u \u2260 u<\/li>\n <li>u < μ<\/li>\n <li>μ > μ<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>p \u2260 p<\/li>\n <li>p < p<\/li>\n <li>p > p<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:16px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>μ1<\/sub> \u2260 μ2<\/sub><\/li>\n <li>μ1<\/sub> < μ2<\/sub><\/li>\n <li>μ1<\/sub> > μ2<\/sub><\/li>\n <\/ul>\n <\/td>\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Test Statistic\n <\/th>\n\n \n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n z =\n \n \n x\u0304 \u2212 \u03bc\n <\/span>\n \n \n \n \u03c3\n <\/span>\n \n \u221an\n <\/span>\n <\/span>\n <\/span>\n <\/span>\n <\/td>\n\n \n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n t =\n \n \n x\u0304 \u2212 \u03bc\n <\/span>\n \n \n \n s\n <\/span>\n \n \u221an\n <\/span>\n <\/span>\n <\/span>\n <\/span>\n <\/td>\n\n \n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:16px;\">\n z =\n \n \n p\u0302 \u2212 p\n <\/span>\n \n \u221a<\/span>\n \n \n \n p(1 \u2212 p)\n <\/span>\n \n n\n <\/span>\n <\/span>\n <\/span>\n<\/span>\n<\/td>\n \n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n z =\n \n \n (x\u03041<\/sub> \u2212 x\u03042<\/sub>) \u2212 (\u03bc1<\/sub> \u2212 \u03bc2<\/sub>)\n <\/span>\n \n \u221a<\/span>\n\n \n \n \u03c31<\/sub>2<\/sup>\n <\/span>\n \n n1<\/sub>\n <\/span>\n <\/span>\n\n +\n\n \n \n \u03c32<\/sub>2<\/sup>\n <\/span>\n \n n2<\/sub>\n <\/span>\n <\/span>\n <\/span>\n<\/span>\n <\/td>\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Requirements\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>n \u2265 30<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>n \u2265 30<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>np \u2265 10<\/li>\n <li>n(1-p) \u2265 10<\/li>\n <li>Approximately Normal<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>n1<\/sub> \u2265 30 and n2<\/sub> \u2265 30<\/li>\n <li>Both Populations Approximately Normal<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Key Phrases\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Test<\/li>\n <li>Claim<\/li>\n <li>Sufficient Evidence<\/li>\n <li>Mean<\/li>\n <li>Population Standard Deviation Known<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Test<\/li>\n <li>Claim<\/li>\n <li>Sufficient Evidence<\/li>\n <li>Mean<\/li>\n <li>Population Standard Deviation Unknown<\/li>\n <li>Sample Standard Deviation<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Test<\/li>\n <li>Claim<\/li>\n <li>Sufficient Evidence<\/li>\n <li>Proportion<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Test<\/li>\n <li>Claim<\/li>\n <li>Sufficient Evidence<\/li>\n <li>Difference Between Two Means<\/li>\n <li>Population Standard Deviation Known<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n“\n<tr>\n <th rowspan=\"2\" style=\"border:1px solid #000; padding:8px;\">\n Conclusions\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n p \u2212 value \u2264 \u03b1<\/em>\n <\/td>\n\n <td colspan=\"4\" style=\"border:1px solid #000; padding:8px;\">\n Reject the null hypothesis. \n There is sufficient evidence to support the alternative hypothesis.\n <\/td>\n<\/tr>\n\n<tr>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">\n p \u2212 value > \u03b1<\/em>\n <\/td>\n\n <td colspan=\"4\" style=\"border:1px solid #000; padding:8px;\">\n Do not reject the null hypothesis. \n There is not sufficient evidence to support the alternative hypothesis.\n <\/td>\n<\/tr>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"5\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Test Calculations\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Z Test\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n T Test\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Proportion Test\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n 2 Sample Z Test\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n General\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>1: Z-Test<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if List (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>2: T-Test<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if List (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>5: 1-PropZTest<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Stat > Tests<\/li>\n <li>3: 2-SampZTest<\/li>\n <li>Stat if Numbers<\/li>\n <li>Data if Lists (Stat > Edit)<\/li>\n <\/ul>\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Calculator Input\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>μ0<\/sub>: Null Hypothesis<\/li>\n <li>σ: Population SD<\/li>\n <li>x\u0304: Sample Mean<\/li>\n <li>n: Sample Size<\/li>\n <li>μ: ≠ μ0<\/sub>, < μ0<\/sub>, > μ0<\/sub> Alternative Hypothesis<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>μ0<\/sub>: Null Hypothesis<\/li>\n <li>x\u0304: Sample Mean<\/li>\n <li>Sx: Sample SD<\/li>\n <li>n: Sample Size<\/li>\n <li>μ: ≠ μ0<\/sub>, < μ0<\/sub>, > μ0<\/sub> Alternative Hypothesis<\/li>\n <\/ul>\n <\/td>\n<td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>p0<\/sub>: Null Hypothesis<\/li>\n <li>x: Number of Successes<\/li>\n <li>n: Sample Size<\/li>\n <li>prop: \u2260 p0<\/sub>, < p0<\/sub>, > p0<\/sub> \n Alternative Hypothesis\n <\/li>\n <\/ul>\n<\/td>\n<td style=\"border:1px solid #000; padding:8px; font-size:12px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>σ1<\/sub>: Population SD Group 1<\/li>\n <li>σ2<\/sub>: Population SD Group 2<\/li>\n <li>x\u03041<\/sub>: Sample Mean Group 1<\/li>\n <li>n1<\/sub>: Sample Size Group 1<\/li>\n <li>x\u03042<\/sub>: Sample Mean Group 2<\/li>\n <li>n2<\/sub>: Sample Size Group 2<\/li>\n <li>μ1<\/sub> \u2260 μ2<\/sub> < μ2<\/sub> > μ2<\/sub> \n Alternative Hypothesis\n <\/li>\n <\/ul>\n<\/td>\n<\/tr>\n<\/table>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center\">Chapter 12<\/h3>\n\n\n\n<table style=\"border-collapse:collapse; width:100%;\">\n\n<tr>\n <th colspan=\"7\" style=\"border:1px solid #000; padding:8px; font-size:24px; text-align:center;\">\n Linear Regression\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\"><\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Explanatory Variable\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Response Variable\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Slope\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n y-intercept\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Correlation Coefficient\n <\/th>\n\n <th style=\"border:1px solid #000; padding:8px; text-align:center;\">\n Coefficient of Determination\n <\/th>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Symbol\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">x<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">y<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">a<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">b<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">r<\/td>\n <td style=\"border:1px solid #000; padding:8px; text-align:center;\">r2<\/sup><\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Purpose\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n Predict changes in y\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n Respond to changes in x\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n For every one x, y changes by b\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n If x = 0, y = a\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n <ul style=\"margin:0; padding-left:18px;\">\n <li>Close to -1 or 1: Strong Relationship<\/li>\n <li>Close to 0: Weak Relationship<\/li>\n <li>Sign Matches Correlation (+ or -)<\/li>\n <\/ul>\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; text-align:center; font-size:14px;\">\n Tells how much of y is predicted by x\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n AKA\n <\/th>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Independent \n • Predictor\n <\/td>\n\n <td style=\"border:.1px solid #000; padding:8px; font-size:14px;\">\n • Dependent \n • Predicted\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Rate of Change\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Starting Value\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Strength and Direction of Relationship\n <\/td>\n\n <td style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Accuracy of Data\n <\/td>\n<\/tr>\n\n<tr>\n <th style=\"border:1px solid #000; padding:8px;\">\n Calculator Function\n <\/th>\n\n <td colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Stat > Edit \n • Input Data \n • Stat > Calc \n • 4: LinReg(ax+b)\n <\/td>\n\n <td colspan=\"3\" style=\"border:1px solid #000; padding:8px; font-size:14px;\">\n • Xlist: List With x Data \n • Ylist: List With y Data \n • FreqList: Blank<\/u><\/li>\n <\/td>\n<\/tr>\n<\/table>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg\"><img loading=\"lazy\" decoding=\"async\" width=\"602\" height=\"455\" src=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg\" alt=\"91ԭƬ Learning Center Logo\" class=\"wp-image-4893\" style=\"width:200px\" srcset=\"https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image.jpeg 602w, https:\/\/www.eastcentral.edu\/learning-center\/wp-content\/uploads\/sites\/27\/2025\/05\/image-300x227.jpeg 300w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/><\/a><figcaption class=\"wp-element-caption\">For more help, please visit The Learning Center<\/figcaption><\/figure>\n<\/div>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"Calculations Chapter 2 Frequency Distributions Consider the following data representing the number of miles professors drive to work each day: 3.8, 2.7, 9.3, 6.5, 5.8, 7, 10.2, 1, 3.7, 9.1, 6.2, 11, 11.9, 5.5, 4.8, 7.3, 9.1, 1.4 Number of Miles Professors Drive to Work Each Day Number of Miles Professors Drive to Work Each […]<\/p>\n","protected":false},"author":12,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_eb_attr":"","footnotes":""},"yst_prominent_words":[],"class_list":["post-6309","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/users\/12"}],"replies":[{"embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/comments?post=6309"}],"version-history":[{"count":131,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6309\/revisions"}],"predecessor-version":[{"id":6526,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6309\/revisions\/6526"}],"wp:attachment":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/media?parent=6309"}],"wp:term":[{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/yst_prominent_words?post=6309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}