8.9: Chapter 8 Formula Review

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8.1 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): \(t_{c}=\frac{\left(\overline{x}_{1}-\overline{x}_{2}\right)-\delta_{0}}{\sqrt{\frac{\left(s_{1}\right)^{2}}{n_{1}}+\frac{\left(s_{2}\right)^{2}}{n_{2}}}}\)

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.

\(\overline{x}_{1}\) and \(\overline{x}_{2}\) are the sample means.

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

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

\(d=\frac{\overline{x}_{1}-\overline{x}_{2}}{s_{\text {pooled}}}\)
where \(s_{\text {pooled}}=\sqrt{\frac{\left(n_{1}-1\right) s_{1}^{2}+\left(n_{2}-1\right) s_{2}^{2}}{n_{1}+n_{2}-2}}\)

8.3 Test for Differences in Means: Assuming Equal Population Variances

\[t_{c}=\frac{\left(\overline{x}_{1}-\overline{x}_{2}\right)-\delta_{0}}{\sqrt{(s_{p}^{2})}}\nonumber\]

where \(s_{p}^{2}\) is the pooled variance given by the formula:

\[s_{p}^{2}=\frac{\left(n_{1}-1\right) s_{2}^{1}+\left(n_{2}-1\right) s_{2}^{2}}{n_{1}+n_{2}-2}\nonumber\]

8.4 Comparing Two Independent Population Proportions

Pooled Proportion: \(p_{c}=\frac{x_{A}+x_{B}}{n_{A}+n_{B}}\)

Test Statistic (z-score): \(Z_{c}=\frac{\left(p^{\prime}_{A}-p^{\prime}_{B}\right)}{\sqrt{p_{c}\left(1-p_{c}\right)\left(\frac{1}{n_{A}}+\frac{1}{n_{B}}\right)}}\)

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.

8.5 Two Population Means with Known Standard Deviations

Test Statistic (z-score):

\(Z_{c}=\frac{\left(\overline{x}_{1}-\overline{x}_{2}\right)-\delta_{0}}{\sqrt{\frac{\left(\sigma_{1}\right)^{2}}{n_{1}}+\frac{\left(\sigma_{2}\right)^{2}}{n_{2}}}}\)

where:
\(\sigma_1\) and \(\sigma_2\) are the known population standard deviations. \(n_1\) and \(n_2\) are the sample sizes. \(\overline{x}_{1}\) and \(\overline{x}_{2}\) are the sample means. \(\mu_1\) and \(\mu_2\) are the population means.

8.6 Matched or Paired Samples

Test Statistic (t-score): \(t_{c}=\frac{\overline{x}_{d}-\mu_{d}}{\left(\frac{s_{d}}{\sqrt{n}}\right)}\)

where:

\(\overline{x}_{d}\) 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.