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Formula Review

Standard error: SE = $\sqrt{\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}}} \sqrt{\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}}}$

Test statistic (t-score): t = $\frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - (μ_{1} - μ_{2})}{\sqrt{\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}}}} \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - (μ_{1} - μ_{2})}{\sqrt{\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}}}}$

Degrees of freedom:
$d f = \frac{{(\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}})}^{2}}{(\frac{1}{n_{1} - 1}) {(\frac{{(s_{1})}^{2}}{n_{1}})}^{2} + (\frac{1}{n_{2} - 1}) {(\frac{{(s_{2})}^{2}}{n_{2}})}^{2}} d f = \frac{{(\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}})}^{2}}{(\frac{1}{n_{1} - 1}) {(\frac{{(s_{1})}^{2}}{n_{1}})}^{2} + (\frac{1}{n_{2} - 1}) {(\frac{{(s_{2})}^{2}}{n_{2}})}^{2}}$

where:

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

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

Cohen’s d is the measure of effect size:

$d = \frac{{\bar{x}}_{1} - {\bar{x}}_{2}}{s_{p o o l e d}} d = \frac{{\bar{x}}_{1} - {\bar{x}}_{2}}{s_{p o o l e d}}$
where $s_{p o o l e d} = \sqrt{\frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}}{n_{1} + n_{2} - 2}} . s_{p o o l e d} = \sqrt{\frac{(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}}{n_{1} + n_{2} - 2}} .$

Normal distribution:
${\bar{X}}_{1} - {\bar{X}}_{2} \sim N [μ_{1} - μ_{2}, \sqrt{\frac{{(σ_{1})}^{2}}{n_{1}} + \frac{{(σ_{2})}^{2}}{n_{2}}}] {\bar{X}}_{1} - {\bar{X}}_{2} \sim N [μ_{1} - μ_{2}, \sqrt{\frac{{(σ_{1})}^{2}}{n_{1}} + \frac{{(σ_{2})}^{2}}{n_{2}}}]$ .
Generally, µ₁ – µ₂ = 0.

Test statistic (z-score):

$z = \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - (μ_{1} - μ_{2})}{\sqrt{\frac{{(σ_{1})}^{2}}{n_{1}} + \frac{{(σ_{2})}^{2}}{n_{2}}}} z = \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - (μ_{1} - μ_{2})}{\sqrt{\frac{{(σ_{1})}^{2}}{n_{1}} + \frac{{(σ_{2})}^{2}}{n_{2}}}}$

Generally, µ₁ - µ₂ = 0.

where
σ₁ and σ₂ are the known population standard deviations, n₁ and n₂ are the sample sizes, ${\bar{x}}_{1} {\bar{x}}_{1}$ and ${\bar{x}}_{2} {\bar{x}}_{2}$ are the sample means, and μ₁ and μ₂ are the population means.

Pooled proportion: p_c = $\frac{x_{F} + x_{M}}{n_{F} + n_{M}} \frac{x_{F} + x_{M}}{n_{F} + n_{M}}$

Distribution for the differences:
${p^{'}}_{A} - {p^{'}}_{B} \sim N [0, \sqrt{p_{c} (1 - p_{c}) (\frac{1}{n_{A}} + \frac{1}{n_{B}})}] {p^{'}}_{A} - {p^{'}}_{B} \sim N [0, \sqrt{p_{c} (1 - p_{c}) (\frac{1}{n_{A}} + \frac{1}{n_{B}})}]$

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

Test statistic (z-score): $z = \frac{(p^{'}_{A} - p^{'}_{B})}{\sqrt{p_{c} (1 - p_{c}) (\frac{1}{n_{A}} + \frac{1}{n_{B}})}} z = \frac{(p^{'} A - p^{'} B)}{\sqrt{p_{c} (1 - p_{c}) (\frac{1}{n_{A}} + \frac{1}{n_{B}})}}$

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.

Test statistic (t-score): t = $\frac{{\bar{x}}_{d} - μ_{d}}{(\frac{s_{d}}{\sqrt{n}})} \frac{{\bar{x}}_{d} - μ_{d}}{(\frac{s_{d}}{\sqrt{n}})}$

where:

${\bar{x}}_{d} {\bar{x}}_{d}$ 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.