{"id":6437,"date":"2026-07-08T14:31:12","date_gmt":"2026-07-08T19:31:12","guid":{"rendered":"https:\/\/www.eastcentral.edu\/learning-center\/?page_id=6437"},"modified":"2026-07-08T15:16:53","modified_gmt":"2026-07-08T20:16:53","slug":"confidence-intervals-hypothesis-testing","status":"publish","type":"page","link":"https:\/\/www.eastcentral.edu\/learning-center\/confidence-intervals-hypothesis-testing\/","title":{"rendered":"Confidence Intervals & Hypothesis Testing"},"content":{"rendered":"\n\n\n\n \n\n \n\n \n\n \n\n \n\n\n\n\n\n\n\n\n\n\n
\n Confidence Intervals\n <\/th>\n<\/tr>\n\n
<\/th>\n\n \n Mean\n <\/th>\n\n \n Proportion\n <\/th>\n\n \n Two Means\n <\/th>\n\n<\/tr>\n\n
\n \u03c3 Known\n <\/th>\n\n \n \u03c3 Unknown\n <\/th>\n\n<\/tr>\n\n
\n Formula\n <\/th>\n\n \n (x\u0304 \u2212 E, x\u0304 + E)\n <\/td>\n\n \n (x\u0304 \u2212 E, x\u0304 + E)\n <\/td>\n\n \n (p\u0302 \u2212 E, p\u0302 + E)\n <\/td>\n\n \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
\n Calculator
Function\n <\/th>\n\n 		\n ZInterval\n <\/td>\n\n \n TInterval\n <\/td>\n\n \n 1-PropZInt\n <\/td>\n\n \n 2-SampZInt\n <\/td>\n\n<\/tr>\n
\n Margin of Error\n<\/th>\n\n\n\nE = z\n\n\n \u03b1<\/span>\n 2<\/span>\n<\/span>\n\n\n \n \u03c3\n <\/span>\n \n \u221an\n <\/span>\n<\/span>\n\n<\/td>\n\n\n\nE = t\n\n\n \u03b1<\/span>\n 2<\/span>\n<\/span>\n\n\n \n s\n <\/span>\n \n \u221an\n <\/span>\n<\/span>\n\n<\/td>\n\n\n\nE = z\n\n\n \u03b1<\/span>\n 2<\/span>\n<\/span>\n\n\n\n \u221a<\/span>\n\n \n\n \n\n \n p\u0302(1 \u2212 p\u0302)\n <\/span>\n\n \n n\n <\/span>\n\n <\/span>\n\n <\/span>\n\n<\/span>\n\n<\/td>\n\n\n\nE = z\n\n\n \u03b1<\/span>\n 2<\/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\n <\/span>\n\n<\/span>\n\n<\/td>\n\n<\/tr>\n
\n Critical Value\n<\/th>\n\n\nz\u03b1\/2<\/sub>\n<\/td>\n\n\nt\u03b1\/2<\/sub>\n<\/td>\n\n\nz\u03b1\/2<\/sub>\n<\/td>\n\n\nz\u03b1\/2<\/sub>\n<\/td>\n\n<\/tr>\n
\n Requirements\n<\/th>\n\n\n\n\u2022 n \u2265 30
\n\u2022 Approximately Normal\n\n<\/td>\n\n		\n\n\u2022 n \u2265 30
\n\u2022 Approximately Normal\n\n<\/td>\n\n		\n\n\u2022 np \u2265 10
\n\u2022 n(1 \u2212 p\u0302) \u2265 10
\n\u2022 Approximately Normal\n\n<\/td>\n\n		\n\n\u2022 n\u2081 \u2265 30 & n\u2082 \u2265 30
\n\u2022 Both Populations Approximately Normal\n\n<\/td>\n\n<\/tr>\n
\n Key Phrases\n<\/th>\n\n\n\n\u2022 Interval
\n\u2022 Endpoints
\n\u2022 Mean
\n\u2022 Population Standard Deviation Known\n\n<\/td>\n\n		\n\n\u2022 Interval
\n\u2022 Endpoints
\n\u2022 Mean
\n\u2022 Population Standard Deviation Unknown
\n\u2022 Sample Standard Deviation\n\n<\/td>\n\n		\n\n\u2022 Interval
\n\u2022 Endpoints
\n\u2022 Proportion\n\n<\/td>\n\n		\n\n\u2022 Interval
\n\u2022 Endpoints
\n\u2022 Difference Between Two Means
\n\u2022 Population Standard Deviation Known\n\n<\/td>\n\n<\/tr>\n
\n Extra Equations\n<\/th>\n\n\n\n\u2022 E =\n\n\n \n upper endpoint \u2212 lower endpoint\n <\/span>\n \n 2\n <\/span>\n<\/span>\n\n

\n\n\u2022 \u03bc =\n\n\n \n upper endpoint + lower endpoint\n <\/span>\n \n 2\n <\/span>\n<\/span>\n\n<\/td>\n\n		\n N\/A\n<\/td>\n\n\n\n\u2022 E =\n\n\n \n upper \u2212 lower\n <\/span>\n \n 2\n <\/span>\n<\/span>\n\n

\n\n\u2022 p =\n\n\n \n upper + lower\n <\/span>\n \n 2\n <\/span>\n<\/span>\n\n

\n\n\u2022 x = p\u0302 × n\n\n<\/td>\n\n		\n\n\u2022 E =\n\n\n \n upper \u2212 lower\n <\/span>\n \n 2\n <\/span>\n<\/span>\n\n

\n\n\u2022 \u03bc\u2081 \u2212 \u03bc\u2082 =\n\n\n \n upper + lower\n <\/span>\n \n 2\n <\/span>\n<\/span>\n\n<\/td>\n\n<\/tr>\n\n\n\n
\n\n\n\n \n \n \n \n \n\n \n \n\n \n\n
\n Critical Value Calculations\n <\/th>\n <\/tr>\n\n
<\/th>\n\n \n Z Interval\n <\/th>\n\n \n T Interval\n <\/th>\n\n \n Proportion Interval\n <\/th>\n\n \n 2 Sample Z Interval\n <\/th>\n <\/tr>\n\n
\n Symbol\n <\/th>\n\n \n z\u03b1\/2<\/sub><\/em>\n <\/td>\n\n \n t\u03b1\/2<\/sub><\/em>\n <\/td>\n\n \n z\u03b1\/2<\/sub><\/em>\n <\/td>\n\n \n z\u03b1\/2<\/sub><\/em>\n <\/td>\n <\/tr>\n <\/thead>\n\n
\n General\n <\/th>\n\n \n \u2022 2nd<\/sup> > Vars
\n \u2022 invNorm\n <\/td>\n\n 		\n \u2022 Stat > Tests
\n \u2022 8: TInterval
\n \u2022 Stat\n <\/td>\n\n 		\n \u2022 2nd<\/sup> > Vars
\n \u2022 invT\n <\/td>\n\n 		\n Same as Z Interval\n <\/td>\n\n \n Same as Z Interval\n <\/td>\n <\/tr>\n\n
\n TI-83\n <\/th>\n\n \n \n 1 \u2212 c<\/span>\n 2<\/span>\n <\/span>\n , 0, 1\n <\/td>\n\n \n \u2022 x\u0304: 0
\n \u2022 Sx: \u221an
\n \u2022 n: Sample Size
\n \u2022 C-Level: c\n <\/td>\n\n 		\n N\/A\n <\/td>\n\n<\/tr>\n\n
\n TI-84\n <\/th>\n\n \n \u2022 Area:\n \n 1 \u2212 c<\/span>\n 2<\/span>\n <\/span>\n
\n \u2022 Mean: 0
\n \u2022 SD: 1\n <\/td>\n\n 		\n \u2022 x\u0304: 0
\n \u2022 Sx: \u221an
\n \u2022 n: Sample Size
\n \u2022 C-Level: c\n <\/td>\n\n 		\n \u2022 Area:\n \n 1 \u2212 c<\/span>\n 2<\/span>\n <\/span>\n
\n \u2022 DF:
n \u2212 1\n <\/td>\n\n<\/tr>\n\n\n\n\n \n \n \n \n
\n Endpoint Calculations\n <\/th>\n <\/tr>\n\n
<\/th>\n Z Interval<\/th>\n T Interval<\/th>\n Proportion Interval<\/th>\n 2 Sample Z Interval<\/th>\n <\/tr>\n\n
General<\/th>\n\n \n \u2022 Stat > Tests
\n \u2022 7: ZInterval
\n \u2022 Stat if Numbers
\n \u2022 Data if List (Stat > Edit)\n <\/td>\n\n 		\n \u2022 Stat > Tests
\n \u2022 8: TInterval
\n \u2022 Stat if Numbers
\n \u2022 Data if List (Stat > Edit)\n <\/td>\n\n 		\n \u2022 Stat > Tests
\n \u2022 A: 1-PropZInt\n <\/td>\n\n 		\n \u2022 Stat > Tests
\n \u2022 9: 2-SampZInt
\n \u2022 Stat if Numbers
\n \u2022 Data if Lists (Stat > Edit)\n <\/td>\n <\/tr>\n\n
Calculator Input<\/th>\n\n \n \u2022 \u03c3: Population SD
\n \u2022 x\u0304: Sample Mean
\n \u2022 n: Sample Size
\n \u2022 C-Level: c\n <\/td>\n\n 		\n \u2022 x\u0304: Sample Mean
\n \u2022 Sx: Sample SD
\n \u2022 n: Sample Size
\n \u2022 C-Level: c\n <\/td>\n\n 		\n \u2022 x: Number of Successes
\n \u2022 n: Sample Size
\n \u2022 C-Level: c\n <\/td>\n\n 		\n \u2022 \u03c31: Population SD Group 1
\n \u2022 \u03c32: Population SD Group 2
\n \u2022 x\u03041: Sample Mean Group 1
\n \u2022 n1: Sample Size Group 1
\n \u2022 x\u03042: Sample Mean Group 2
\n \u2022 n2: Sample Size Group 2
\n \u2022 C-Level: c\n <\/td>\n <\/tr>\n<\/table>\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n
\n Hypothesis Tests\n <\/th>\n<\/tr>\n\n
<\/th>\n Mean<\/th>\n \n Proportion\n <\/th>\n \n Two Means\n <\/th>\n<\/tr>\n\n
\u03c3 Known<\/th>\n \u03c3 Unknown<\/th>\n<\/tr>\n\n
Calculator Function<\/th>\n Z-Test<\/td>\n T-Test<\/td>\n 1-PropZTest<\/td>\n 2-SampZTest<\/td>\n<\/tr>\n\n
Alternative Hypotheses<\/th>\n\n \n \u2022 \u03bc \u2260 \u03bc
\n \u2022 \u03bc < \u03bc
\n \u2022 \u03bc > \u03bc\n <\/td>\n\n 		\n \u2022 \u03bc \u2260 \u03bc
\n \u2022 \u03bc < \u03bc
\n \u2022 \u03bc > \u03bc\n <\/td>\n\n 		\n \u2022 p \u2260 p
\n \u2022 p < p
\n \u2022 p > p\n <\/td>\n\n 		\n \u2022 \u03bc\u2081 \u2260 \u03bc\u2082
\n \u2022 \u03bc\u2081 < \u03bc\u2082
\n \u2022 \u03bc\u2081 > \u03bc\u2082\n <\/td>\n<\/tr>\n\n
Test Statistic<\/th>\n\n \n \n z =\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<\/span>\n<\/td>\n\n \n t =\n \n \n x\u0304 \u2212 \u03bc\n <\/span>\n \n (\n \n \n S\n <\/span>\n \n \u221an\n <\/span>\n <\/span>\n )\n <\/span>\n <\/span>\n<\/td>\n\n \n z =\n \n \n p\u0302 \u2212 p\n<\/span>\n \n \n \u221a\n \n \n \n p(1\u2212p)\n <\/span>\n \n n\n <\/span>\n <\/span>\n <\/span>\n <\/span>\n <\/span>\n<\/span>\n \n \n<\/td>\n\n \n z =\n \n \n \n (x\u0304\u2081 \u2212 x\u0304\u2082) \u2212 (\u03bc\u2081 \u2212 \u03bc\u2082)\n <\/span>\n\n \n \u221a\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<\/td>\n<\/tr>\n\n
Requirements<\/th>\n\n \n \u2022 n \u2265 30
\n \u2022 Approximately Normal\n <\/td>\n\n 		\n \u2022 n \u2265 30
\n \u2022 Approximately Normal\n <\/td>\n\n 		\n \u2022 np \u2265 10
\n \u2022 n(1-p) \u2265 10
\n\u2022Approximately Normal\n <\/td>\n\n 		\n \u2022 n\u2081 \u2265 30 & n\u2082 \u2265 30
\n \u2022 Both Populations Approximately Normal\n <\/td>\n<\/tr>\n\n
Key Phrases<\/th>\n\n \n \u2022 Test
\n \u2022 Claim
\n \u2022 Sufficient Evidence
\n \u2022 Mean
\n \u2022 Population Standard Deviation Known\n <\/td>\n\n 		\n \u2022 Test
\n \u2022 Claim
\n \u2022 Sufficient Evidence
\n \u2022 Mean
\n \u2022 Population Standard Deviation Unknown
\n \u2022 Sample Standard Deviation\n <\/td>\n\n 		\n \u2022 Test
\n \u2022 Claim
\n \u2022 Sufficient Evidence
\n \u2022 Proportion\n <\/td>\n\n 		\n \u2022 Test
\n \u2022 Claim
\n \u2022 Sufficient Evidence
\n \u2022 Difference Between Two Means
\n \u2022 Population Standard Deviation Known\n <\/td>\n<\/tr>\n\n
Conclusions<\/th>\n\n \n p-value \u2264 \u03b1\n <\/td>\n\n \n Reject the null hypothesis.
\n There is sufficient evidence to support the alternative hypothesis.\n <\/td>\n<\/tr>\n\n
\n p-value > \u03b1\n <\/td>\n\n \n Do not reject the null hypothesis.
\n There is not sufficient evidence to support the alternative hypothesis.\n <\/td>\n<\/tr>\n\n<\/table>\n\n\n\n\n\n\n \n \n \n
\n Test Calculations\n <\/th>\n<\/tr>\n\n
<\/th>\n Z Test<\/th>\n T Test<\/th>\n Proportion Test<\/th>\n 2 Sample Z Test<\/th>\n<\/tr>\n\n
General<\/th>\n\n \n \u2022 Stat > Tests
\n \u2022 1: Z-Test
\n \u2022 Stat if Numbers
\n \u2022 Data if List (Stat > Edit)\n <\/td>\n\n 		\n \u2022 Stat > Tests
\n \u2022 2: T-Test
\n \u2022 Stat if Numbers
\n \u2022 Data if List (Stat > Edit)\n <\/td>\n\n 		\n \u2022 Stat > Tests
\n \u2022 5: 1-PropZTest\n <\/td>\n\n 		\n \u2022 Stat > Tests
\n \u2022 3: 2-SampZTest
\n \u2022 Stat if Numbers
\n \u2022 Data if Lists (Stat > Edit)\n <\/td>\n<\/tr>\n\n
\n Calculator Input\n <\/th>\n\n \n \u2022 \u03bc\u2080: Null Hypothesis
\n \u2022 \u03c3: Population SD
\n \u2022 x\u0304: Sample Mean
\n \u2022 n: Sample Size
\n \u2022 \u03bc: \u2260 \u03bc\u2080 < \u03bc\u2080 > \u03bc\u2080
\n    Alternative Hypothesis\n <\/td>\n\n 		\n \u2022 \u03bc\u2080: Null Hypothesis
\n \u2022 x\u0304: Sample Mean
\n \u2022 Sx: Sample SD
\n \u2022 n: Sample Size
\n \u2022 \u03bc: \u2260 \u03bc\u2080 < \u03bc\u2080 > \u03bc\u2080
\n    Alternative Hypothesis\n <\/td>\n\n 		\n \u2022 p\u2080: Null Hypothesis
\n \u2022 x: Number of Successes
\n \u2022 n: Sample Size
\n \u2022 prop: \u2260 p\u2080 < p\u2080 > p\u2080
\n    Alternative Hypothesis\n <\/td>\n\n 		\n \u2022 \u03c31: Population SD Group 1
\n \u2022 \u03c32: Population SD Group 2
\n \u2022 x\u03041: Sample Mean Group 1
\n \u2022 n1: Sample Size Group 1
\n \u2022 x\u03042: Sample Mean Group 2
\n \u2022 n2: Sample Size Group 2
\n \u2022 \u03bc1: \u2260 \u03bc2 < \u03bc2 > \u03bc2
\n    Alternative Hypothesis\n <\/td>\n<\/tr>\n\n<\/table>\n\n\n
\n
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For more help, please contact The Learning Center<\/figcaption><\/figure>\n<\/div>","protected":false},"excerpt":{"rendered":"

Confidence Intervals Mean Proportion Two Means \u03c3 Known \u03c3 Unknown Formula (x\u0304 \u2212 E, x\u0304 + E) (x\u0304 \u2212 E, x\u0304 + E) (p\u0302 \u2212 E, p\u0302 + E) ((x\u0304\u2081 \u2212 x\u0304\u2082) \u2212 E, (x\u0304\u2081 \u2212 x\u0304\u2082) + E) CalculatorFunction ZInterval TInterval 1-PropZInt 2-SampZInt Margin of Error E = z \u03b1 2 \u03c3 \u221an E […]<\/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-6437","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6437","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=6437"}],"version-history":[{"count":9,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6437\/revisions"}],"predecessor-version":[{"id":6451,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/pages\/6437\/revisions\/6451"}],"wp:attachment":[{"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/media?parent=6437"}],"wp:term":[{"taxonomy":"yst_prominent_words","embeddable":true,"href":"https:\/\/www.eastcentral.edu\/learning-center\/wp-json\/wp\/v2\/yst_prominent_words?post=6437"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}