10.9: Formula Review

10.2 Comparing Two Independent Population Means

Standard error: \(S E=\sqrt{\frac{\left(s_1\right)^2}{n_1}+\frac{\left(s_2\right)^2}{n_2}}\)

Test statistic ( \(t\)-score): \(t_c=\frac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{\frac{\left(s_1\right)^2}{n_1}+\frac{\left(s_2\right)^2}{n_2}}}\)

Degrees of freedom:
\[d f=\frac{\left(\frac{\left(s_1\right)^2}{n_1}+\frac{\left(s_2\right)^2}{n_2}\right)^2}{\left(\frac{1}{n_1-1}\right)\left(\frac{\left(s_1\right)^2{n_1}\right)^2+\left(\frac{1}{n_2-1}\right)\left(\frac{\left(s_2\right)^2}{n_2}\right)^2}\]

where:

\(s_1\) and \(s_2\) are the sample standard deviations, and \(n_1\) and \(n_2\) are the sample sizes.

\(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.

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

Cohen's \(d\) is the measure of effect size:

\[d=\frac{\bar{x}_1-\bar{x}_2}{s_{\text {pooled }}}\]

where \(s_{\text {pooled }}=\sqrt{\frac{\left(n_1-1\right) s_1^2+\left(n_2-1\right) s_2^2}{n_1+n_2-2}}\)

10.4 Test for Differences in Means: Assuming Equal Population Variances

\[t_c=\frac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{S_p^2\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}\]

where \(S p\) is the pooled variance given by the formula:

\[S p=\frac{\left(n_1-1\right) s_1^2-\left(n_2-1\right) s_2^2}{n_1+n_2-2}\]

10.5 Comparing Two Independent Population Proportions

Pooled Proportion: \(p_C=\frac{x_A+x_B}{n_A+n_B}\)

Test Statistic (z-score): \(Z_c=\frac{\left(p^{\prime}{ }_A-p_B^{\prime} B\right)}{\sqrt{p_c\left(1-p_c\right)\left(\frac{1}{n_A}+\frac{1}{n_B}\right)}}\)

where

\(p_A^{\prime}\) and \(p_B^{\prime}\) 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.

10.6 Two Population Means with Known Standard Deviations

Test Statistic ( \(z\)-score):

\[Z_c=\frac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{\frac{\left(\sigma_1\right)^2}{n_1}+\frac{\left(\sigma_2\right)^2}{n_2}}\]

where:

\(\sigma_1\) and \(\sigma_2\) are the known population standard deviations. \(n_1\) and \(n_2\) are the sample sizes. \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means. \(\mu_1\) and \(\mu_2\) are the population means.

10.7 Matched or Paired Samples

Test Statistic ( \(t\)-score): \(t_c=\frac{\bar{x}_d-\mu_d}{\left(\frac{s_d}{\sqrt{n}}\right)}\)

where:

\(\bar{x}_d\) is the mean of the sample differences. \(\mu_{\mathrm{d}}\) is the mean of the population differences. \(s_d\) is the sample standard deviation of the differences. \(n\) is the sample size.