# Formula Review

### 10.1Comparing Two Independent Population Means

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): tc = $( x ¯ 1 − x ¯ 2 )−δ0 ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2 ( x ¯ 1 − x ¯ 2 )−δ0 ( s 1 ) 2 n 1 + ( s 2 ) 2 n 2$

Degrees of freedom:

where:

$s1s1$ and $s2s2$ are the sample standard deviations, and $n1n1$ and $n2n2$ are the sample sizes.

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

### 10.2Cohen's Standards for Small, Medium, and Large Effect Sizes

Cohen’s d is the measure of effect size:

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

### 10.3Test for Differences in Means: Assuming Equal Population Variances

$tc=(x¯1−x¯2)−δ0Sp2(1n1+1n2)tc=(x¯1−x¯2)−δ0Sp2(1n1+1n2)$

where $Sp2Sp2$ is the pooled variance given by the formula:

$Sp2=(n1−1)s21+(n2−1)s22n1+n2−2Sp2=(n1−1)s21+(n2−1)s22n1+n2−2$

### 10.4Comparing Two Independent Population Proportions

Pooled Proportion: pc =

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

where

$pA'pA'$ and $pB'pB'$ are the sample proportions, $pApA$ and $pBpB$are the population proportions,

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

### 10.5Two Population Means with Known Standard Deviations

Test Statistic (z-score):

$Zc= ( x – 1 − x – 2 )−δ0 ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2 Zc= ( x – 1 − x – 2 )−δ0 ( σ 1 ) 2 n 1 + ( σ 2 ) 2 n 2$

where:
$σ1σ1$ and $σ2σ2$ 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.

### 10.6Matched or Paired Samples

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

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.