Normal Distributions
Normal Probability
- Find
-
- Area under the curve, probability, percentile, percent
- Standard Normal Distribution
-
- Mean = 0 SD = 1
| Conditions | Probability | Calculator Input | Key Phrases | |
|---|---|---|---|---|
|
• Continuous Data • Symmetrical • Mean in Middle • Data Separated by Standard Deviation • Area Adds to 1 |
• P(X < x) • P(X ≤ x) |
• 2nd > Vars • 2: normalcdf |
• Lower: -1E99 • Upper: X • Mean: μ • SD: σ |
• Less Than • At Most • Fewer Than • X or Less |
|
• P(X > x) • P(X ≥ x) |
• Lower: X • Upper: 1E99 • Mean: μ • SD: σ |
• Greater Than • At Least • More Than • X or More |
||
|
• P(X ≤ x ≤ Y) • P(X < x < Y) |
• Lower: X • Upper: Y • Mean: μ • SD: σ |
• Between | ||
|
• P(x < X or x > Y) • P(x ≤ X or x ≥ Y) |
1 − “Between” |
• Tails • Less than X or Greater than Y |
||
| Differs by Less |
• Lower: μ − X • Upper: μ + X • Mean: μ • SD: σ |
• Differs by Less Than X • Differs by Fewer Than X |
||
| Differs by More |
• 1 – “Differs by Less” • 2 * normalcdf a tail |
• Differs by Greater Than X • Differs by More Than X |
||
invNorm
- Find
-
- z-score
-
- Measurements
| Conditions | Direction | Calculator Input | Key Phrases | ||
|---|---|---|---|---|---|
| TI-83 | TI-84 | ||||
|
• Continuous Data • Symmetrical • Mean in Middle • Data Separated by Standard Deviation • Area Adds to 1 |
Left |
• 2nd > Vars • 3: invNorm |
p, μ, σ |
• Area: p • Mean: μ • SD: σ • Left |
• To the Left • Bottom % • Percentile • Less Than |
| Right | (1-p), μ, σ |
• Area: p • Mean: μ • SD: σ • Right |
• To the Right • Top % • More Than |
||
|
Between -z and z |
•
1-p
2
, 0,1 Gives -z • z = -z × -1 |
• Area: p • Mean: 0 • SD: 1 • Center |
• Between • This is ONLY z-scores |
||
| Left -z and Right z |
•
p
2
, 0,1 Gives -z • z = -z × -1 |
• Area: 1-p • Mean: 0 • SD: 1 • Center |
• Left of -z Plus Right of z • This is ONLY z-scores |
||
• Approximating a Binomial Distribution
o MUST follow the binomial distribution rules:
▪ Fixed number of trials: n
▪ Independent trials
▪ Two outcomes: success and failure
▪ Probability for success is the same every time: p
▪ 𝑛𝑝 ≥ 10 𝑛(1−𝑝)≥10
o 𝜇= 𝑛𝑝
o σ = √ np(1 − p)
o MUST use continuity corrections
▪ After the continuity correction, this is the same as normalcdf
| Examples of Continuity Correction | |||
|---|---|---|---|
| Statement | Symbol | Graph | Area |
|
• At least 45 • 45 or more • No fewer than 45 |
≥45 |
|
Right of 44.5 |
|
• More than 45 • Greater than 45 |
>45 |
| Right of 45.5 |
|
• At most 45 • No more than 45 |
≤45 |
|
Left of 45.5 |
|
• Less than 45 • Fewer than 45 |
<45 |
|
Left of 44.5 |
|
• Exactly 45 • Equal to 45 |
=45 |
|
Between 44.5 and 45.5 |
