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Statistics

Formula Review

StatisticsFormula Review

10.1 Two 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:
 df=  ( ( s 1 ) 2 n 1 +  ( s 2 ) 2 n 2 ) 2 ( 1 n 1 −1 ) ( ( s 1 ) 2 n 1 ) 2 +( 1 n 2 −1 ) ( ( s 2 ) 2 n 2 ) 2 df=  ( ( s 1 ) 2 n 1 +  ( s 2 ) 2 n 2 ) 2 ( 1 n 1 −1 ) ( ( s 1 ) 2 n 1 ) 2 +( 1 n 2 −1 ) ( ( s 2 ) 2 n 2 ) 2

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 .

10.2 Two 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, µ1 – µ2 = 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.

10.3 Comparing Two Independent Population Proportions

Pooled proportion: pc = x F  +  x M n F  +  n M x F  +  x M n F  +  n M

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: pA – pB = 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: pA − pB = 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.

10.4 Matched 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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