### 10.1 Comparing Two Independent Population Means

Standard error: *SE* = $\sqrt{\frac{{({s}_{1})}^{2}}{{n}_{1}}+\frac{{({s}_{2})}^{2}}{{n}_{2}}}$

Test statistic (*t*-score): *t _{c}* = $\frac{({\overline{x}}_{1}-{\overline{x}}_{2})-{\delta}_{0}}{\sqrt{\frac{{({s}_{1})}^{2}}{{n}_{1}}+\frac{{({s}_{2})}^{2}}{{n}_{2}}}}$

Degrees of freedom:

$df=\frac{{\left(\frac{{({s}_{1})}^{2}}{{n}_{1}}+\frac{{({s}_{2})}^{2}}{{n}_{2}}\right)}^{2}}{\left(\frac{1}{{n}_{1}-1}\right){\left(\frac{{({s}_{1})}^{2}}{{n}_{1}}\right)}^{2}+\left(\frac{1}{{n}_{2}-1}\right){\left(\frac{{({s}_{2})}^{2}}{{n}_{2}}\right)}^{2}}$

where:

${s}_{1}$ and ${s}_{2}$ are the sample standard deviations, and ${n}_{1}$ and ${n}_{2}$ are the sample sizes.

${\overline{x}}_{1}$ and ${\overline{x}}_{2}$ are the sample means.

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

Cohen’s *d* is the measure of effect size:

$d=\frac{{\overline{x}}_{1}-{\overline{x}}_{2}}{{s}_{pooled}}$

where ${s}_{pooled}=\sqrt{\frac{({n}_{1}-1){s}_{1}^{2}+({n}_{2}-1){s}_{2}^{2}}{{n}_{1}+{n}_{2}-2}}$

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

where $S\stackrel{2}{p}$ is the pooled variance given by the formula:

### 10.4 Comparing Two Independent Population Proportions

Pooled Proportion: *p _{c}* = $\frac{{x}_{A}\text{}+\text{}{x}_{B}}{{n}_{A}\text{}+\text{}{n}_{B}}$

Test Statistic (*z*-score): ${Z}_{c}=\frac{({p}^{\prime}{}_{A}-{p}^{\prime}{}_{B})}{\sqrt{{p}_{c}(1-{p}_{c})\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}}$

where

${p}_{A}^{\text{'}}$ and ${p}_{B}^{\text{'}}$ are the sample proportions, ${p}_{A}$ and ${p}_{B}$are the population proportions,

*P _{c}* is the pooled proportion, and

*n*and

_{A}*n*are the sample sizes.

_{B}### 10.5 Two Population Means with Known Standard Deviations

Test Statistic (*z*-score):

${Z}_{c}=\frac{({\stackrel{\u2013}{x}}_{1}-{\stackrel{\u2013}{x}}_{2})-{\delta}_{0}}{\sqrt{\frac{{({\sigma}_{1})}^{2}}{{n}_{1}}+\frac{{({\sigma}_{2})}^{2}}{{n}_{2}}}}$

**where:**

${\sigma}_{1}$ and ${\sigma}_{2}$ are the known population standard deviations. *n*_{1} and *n*_{2} are the sample sizes. ${\stackrel{\u2013}{x}}_{1}$ and ${\stackrel{\u2013}{x}}_{2}$ are the sample means. *μ*_{1} and *μ*_{2} are the population means.

### 10.6 Matched or Paired Samples

Test Statistic (*t*-score): *t _{c}* = $\frac{{\stackrel{\u2013}{x}}_{d}-{\mu}_{d}}{\left(\frac{{s}_{d}}{\sqrt{n}}\right)}$

where:

${\stackrel{\u2013}{x}}_{d}$ is the mean of the sample differences. *μ*_{d} is the mean of the population differences. *s _{d}* is the sample standard deviation of the differences.

*n*is the sample size.