Skip to Content
Introductory Statistics

# Formula Review

### 10.1Two Population Means with Unknown Standard Deviations

Standard error: SE = $( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2$

Test statistic (t-score): t = $( x ¯ 1 − x ¯ 2 )−( μ 1 − μ 2 ) ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ( x ¯ 1 − x ¯ 2 )−( μ 1 − μ 2 ) ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2$

Degrees of freedom:

where:

s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

$x ¯ 1 x ¯ 1$ and $x ¯ 2 x ¯ 2$ are the sample means.

Cohen’s d is the measure of effect size:

$d= x ¯ 1 − x ¯ 2 s pooled d= x ¯ 1 − x ¯ 2 s pooled$
where $s pooled = ( n 1 −1) s 1 2 +( n 2 −1) s 2 2 n 1 + n 2 −2 s pooled = ( n 1 −1) s 1 2 +( n 2 −1) s 2 2 n 1 + n 2 −2$

### 10.2Two Population Means with Known Standard Deviations

Normal Distribution:
$X ¯ 1 − X ¯ 2 ∼N[ μ 1 − μ 2 , ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ] X ¯ 1 − X ¯ 2 ∼N[ μ 1 − μ 2 , ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 ]$.
Generally µ1µ2 = 0.

Test Statistic (z-score):

$z= ( x ¯ 1 − x ¯ 2 )−( μ 1 − μ 2 ) ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 z= ( x ¯ 1 − x ¯ 2 )−( μ 1 − μ 2 ) ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2$

Generally µ1 - µ2 = 0.

where:
σ1 and σ2 are the known population standard deviations. n1 and n2 are the sample sizes. $x ¯ 1 x ¯ 1$ and $x ¯ 2 x ¯ 2$ are the sample means. μ1 and μ2 are the population means.

### 10.3Comparing Two Independent Population Proportions

Pooled Proportion: pc =

Distribution for the differences:
$p ′ A − p ′ B ∼N[ 0, p c (1− p c )( 1 n A + 1 n B ) ] p ′ A − p ′ B ∼N[ 0, p c (1− p c )( 1 n A + 1 n B ) ]$

where the null hypothesis is H0: pA = pB or H0: pApB = 0.

Test Statistic (z-score): $z= ( p ′ A − p ′ B ) p c (1− p c )( 1 n A + 1 n B ) z= ( p ′ A − p ′ B ) p c (1− p c )( 1 n A + 1 n B )$

where the null hypothesis is H0: pA = pB or H0: pApB = 0.

where

p′A and p′B are the sample proportions, pA and pB are the population proportions,

Pc is the pooled proportion, and nA and nB are the sample sizes.

### 10.4Matched or Paired Samples

Test Statistic (t-score): t = $x ¯ d − μ d ( s d n ) x ¯ d − μ d ( s d n )$

where:

$x ¯ d x ¯ d$ is the mean of the sample differences. μd is the mean of the population differences. sd is the sample standard deviation of the differences. n is the sample size.

Citation/Attribution

Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4.0 and you must attribute OpenStax.

Attribution information
• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction
Citation information

© Sep 19, 2013 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License 4.0 license. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.