10.2 Comparing Two Independent Population Means
Standard error: \(S E=\sqrt{\dfrac{\left(s_1\right)^2}{n_1}+\dfrac{\left(s_2\right)^2}{n_2}}\)
Test statistic ( \(t\)-score): \(t_c=\dfrac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{\dfrac{\left(s_1\right)^2}{n_1}+\dfrac{\left(s_2\right)^2}{n_2}}}\)
Degrees of freedom:
\[d f=\dfrac{\left(\dfrac{\left(s_1\right)^2}{n_1}+\dfrac{\left(s_2\right)^2}{n_2}\right)^2}{\left(\dfrac{1}{n_1-1}\right)\left(\dfrac{\left(s_1\right)^2}{n_1}\right)^2+\left(\dfrac{1}{n_2-1}\right)\left(\dfrac{\left(s_2\right)^2}{n_2}\right)^2}\]
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:
\[d=\dfrac{\bar{x}_1-\bar{x}_2}{s_{\text {pooled }}}\]
where \(s_{\text {pooled }}=\sqrt{\dfrac{\left(n_1-1\right) s_1^2+\left(n_2-1\right) s_2^2}{n_1+n_2-2}}\)
10.4 Test for Differences in Means: Assuming Equal Population Variances
\[t_c=\dfrac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{S_p^2\left(\dfrac{1}{n_1}+\dfrac{1}{n_2}\right)}}\]
where \(S_p^2\) is the pooled variance given by the formula:
\[S_p^2=\dfrac{\left(n_1-1\right) s_1^2+\left(n_2-1\right) s_2^2}{n_1+n_2-2}\]
10.5 Comparing Two Independent Population Proportions
Pooled Proportion: \(p_{\mathrm{C}}=\dfrac{x_A+x_B}{n_A+n_B}\)
Test Statistic (z-score): \(Z_c=\dfrac{\left(p_A^{\prime}-p_B^{\prime}\right)}{\sqrt{p_c\left(1-p_c\right)\left(\dfrac{1}{n_A}+\dfrac{1}{n_B}\right)}}\)
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):
\[Z_c=\dfrac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{\dfrac{\left(\sigma_1\right)^2}{n_1}+\dfrac{\left(\sigma_2\right)^2}{n_2}}}\]
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): \(t_c=\dfrac{\bar{x}_d-\mu_d}{\left(\dfrac{s_d}{\sqrt{n}}\right)}\)
where:
\(\bar{x}_d\) is the mean of the sample differences. \(\mu_d\) is the mean of the population differences. \(s_d\) is the sample standard deviation of the differences. \(n\) is the sample size.