Sections
Formula Review

# Formula Review

### 10.1Two Population Means with Unknown Standard Deviations

Standard error SE = $(s1)2n1+(s2)2n2(s1)2n1+(s2)2n2$

Test statistic (t-score) t = $(x¯1−x¯2)−(μ1−μ2)(s1)2n1+(s2)2n2(x¯1−x¯2)−(μ1−μ2)(s1)2n1+(s2)2n2$

Degrees of freedom

where

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

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

Cohen’s d is the measure of effect size

$d=x¯1−x¯2spooledd=x¯1−x¯2spooled$

where $spooled=(n1−1)s12+(n2−1)s22n1+n2−2.spooled=(n1−1)s12+(n2−1)s22n1+n2−2.$

### 10.2Two Population Means with Known Standard Deviations

Normal distribution

$X¯1−X¯2∼N[μ1−μ2,(σ1)2n1+(σ2)2n2]X¯1−X¯2∼N[μ1−μ2,(σ1)2n1+(σ2)2n2]$.

Generally µ1µ2 = 0.

Test statistic (z-score)

$z=(x¯1−x¯2)−(μ1−μ2)(σ1)2n1+(σ2)2n2z=(x¯1−x¯2)−(μ1−μ2)(σ1)2n1+(σ2)2n2$

Generally µ1 - µ2 = 0

where

σ1 and σ2 are the known population standard deviations, n1 and n2 are the sample sizes, $x¯1x¯1$ and $x¯2x¯2$ 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

$p′A−p′B∼N[0,pc(1−pc)(1nA+1nB)]p′A−p′B∼N[0,pc(1−pc)(1nA+1nB)]$

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

Test statistic (z-score): $z=(p′A−p′B)pc(1−pc)(1nA+1nB)z=(p′A−p′B)pc(1−pc)(1nA+1nB)$

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 = $x¯d−μd(sdn)x¯d−μd(sdn)$

where

$x¯dx¯d$ 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.