# 10.9: Chapter 10 Formula Review

### 10.1 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.

### 10.2 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 $$s_{\text {pooled}}=\sqrt{\frac{\left(n_{1}-1\right) s_{1}^{2}+\left(n_{2}-1\right) s_{2}^{2}}{n_{1}+n_{2}-2}}$$

### 10.3 Test for Differences in Means: Assuming Equal Population Variances

$t_{c}=\frac{\left(\overline{x}_{1}-\overline{x}_{2}\right)-\delta_{0}}{\sqrt{S^{2}\left(\frac{1}{n_{1}}+\frac{1}{n_{2}}\right)}}\nonumber$

where $$S_{p}^{2}$$ is the pooled variance given by the formula:

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

10.4 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.

### 10.5 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.

### 10.6 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.