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

where

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

${\overline{x}}_1$ and ${\overline{x}}_2$ are the sample means.

Cohen’s d is the measure of effect size

$d=\frac{{\overline{x}}_1-{\overline{x}}_2}{s_{pooled}}$

10.2 Two Population Means with Known Standard Deviations

Normal distribution

Test statistic (z-score)

where

10.3 Comparing Two Independent Population Proportions

Pooled proportion p_c = $\frac{x_F+x_M}{n_F+n_M}$

Distribution for the differences

where the null hypothesis is H₀: p_A = p_B or H₀: p_A – p_B = 0.

Test statistic (z-score): $z=\frac{(p^{\prime }{}_A-p^{\prime }{}_B)}{\sqrt{p_c(\mathrm{}\mathrm{}1-p_c)\left(\frac{1}{n_A}+\frac{1}{n_B}\right)}}$

where the null hypothesis is H₀: p_A = p_B or H₀: p_A − p_B = 0

and where

p′_A and p′_B are the sample proportions, p_A and p_B are the population proportions,

P_c is the pooled proportion, and n_A and n_B are the sample sizes.

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.