### 10.1 Two Population Means with Unknown Standard Deviations

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

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

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.

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

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

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

### 10.2 Two Population Means with Known Standard Deviations

Normal distribution:

${\overline{X}}_{1}-{\overline{X}}_{2}\sim N\left[{\mu}_{1}-{\mu}_{2},\sqrt{\frac{{({\sigma}_{1})}^{2}}{{n}_{1}}+\frac{{({\sigma}_{2})}^{2}}{{n}_{2}}}\right]$.

Generally, *µ*_{1} – *µ*_{2} = 0.

Test statistic (*z*-score):

$z=\frac{({\overline{x}}_{1}-{\overline{x}}_{2})-({\mu}_{1}-{\mu}_{2})}{\sqrt{\frac{{({\sigma}_{1})}^{2}}{{n}_{1}}+\frac{{({\sigma}_{2})}^{2}}{{n}_{2}}}}$

Generally, *µ*_{1} - *µ*_{2} = 0.

where

*σ*_{1} and *σ*_{2} are the known population standard deviations, *n*_{1} and *n*_{2} are the sample sizes, ${\overline{x}}_{1}$ and ${\overline{x}}_{2}$ are the sample means, and *μ*_{1} and *μ*_{2} are the population means.

### 10.3 Comparing Two Independent Population Proportions

Pooled proportion: *p _{c}* = $\frac{{x}_{F}\text{}+\text{}{x}_{M}}{{n}_{F}\text{}+\text{}{n}_{M}}$

Distribution for the differences:

$${{p}^{\prime}}_{A}-{{p}^{\prime}}_{B}\sim N\left[0,\sqrt{{p}_{c}(1-{p}_{c})\left(\frac{1}{{n}_{A}}+\frac{1}{{n}_{B}}\right)}\right]$$

where the null hypothesis is *H _{0}*:

*p*=

_{A}*p*or

_{B}*H*:

_{0}*p*–

_{A}*p*= 0

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

where the null hypothesis is *H _{0}*:

*p*=

_{A}*p*or

_{B}*H*:

_{0}*p*−

_{A}*p*= 0

_{B}and where

*p′ _{A}*
and

*p′*are the sample proportions,

_{B}*p*and

_{A}*p*are the population proportions,

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

*n*and

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

_{B}### 10.4 Matched or Paired Samples (Optional)

Test statistic (*t*-score): *t* = $\frac{{\overline{x}}_{d}-{\mu}_{d}}{\left(\frac{{s}_{d}}{\sqrt{n}}\right)}$

where:

${\overline{x}}_{d}$ 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, and

*n*is the sample size.