8.8: Chapter 8 Formula Review

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8.2 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): \(t_{obs}=\frac{\left(\overline{x}_{1}-\overline{x}_{2}\right)-\left(\mu_{1}-\mu_{2}\right)}{\sqrt{\frac{\left(s_{1}\right)^{2}}{n_{1}}+\frac{\left(s_{2}\right)^{2}}{n_{2}}}}\)

Degrees of freedom: \(d f = {n}_{1}+{n}_{2}-2\)

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

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

\(d=\frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{s_{1}^{2}}{2}+\frac{s_{2}^{2}}{2}}}\)

8.4 Comparing Two Independent Population Proportions

Confidence Interval: \(\left(P_{2}^{\prime}-P_{1}^{\prime}\right) \pm z_\frac{\alpha}{2} * \sqrt{\frac{P_{1}^{\prime} *\left(1-P_{1}^{\prime}\right)}{n_{1}}+\frac{P_{2}^{\prime} *\left(1-P_{2}^{\prime}\right)}{n_{2}}}\)

8.5 Matched or Paired Samples

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

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, and \(n\) is the sample size.