Statistics

# Formula Review

StatisticsFormula 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, and 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

and 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 (Optional)

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, and n is the sample size.

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