# 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.

8.8: Chapter 8 Formula Review is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.