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# 10.10: Formula Review

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## 10.2 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): $$t_c=\frac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{\frac{\left(s_1\right)^2}{n_1}+\frac{\left(s_2\right)^2}{n_2}}}$$

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.

$$\bar{x}_1$$ 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:

$d=\frac{\bar{x}_1-\bar{x}_2}{s_{\text {pooled }}}$

where $$s_{\text {pooled }}=\sqrt{\frac{\left(n_1-1\right) s_1^2+\left(n_2-1\right) s_2^2}{n_1+n_2-2}}$$

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

$t_c=\frac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{S_p^2\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}$

where $$S p$$ is the pooled variance given by the formula:

$S p=\frac{\left(n_1-1\right) s_1^2-\left(n_2-1\right) s_2^2}{n_1+n_2-2}$

## 10.5 Comparing Two Independent Population Proportions

Pooled Proportion: $$p_C=\frac{x_A+x_B}{n_A+n_B}$$

Test Statistic (z-score): $$Z_c=\frac{\left(p^{\prime}{ }_A-p_B^{\prime} B\right)}{\sqrt{p_c\left(1-p_c\right)\left(\frac{1}{n_A}+\frac{1}{n_B}\right)}}$$

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.

## 10.6 Two Population Means with Known Standard Deviations

Test Statistic ( $$z$$-score):

$Z_c=\frac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{\frac{\left(\sigma_1\right)^2}{n_1}+\frac{\left(\sigma_2\right)^2}{n_2}}$

where:

$$\sigma_1$$ and $$\sigma_2$$ are the known population standard deviations. $$n_1$$ and $$n_2$$ are the sample sizes. $$\bar{x}_1$$ 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): $$t_c=\frac{\bar{x}_d-\mu_d}{\left(\frac{s_d}{\sqrt{n}}\right)}$$

where:

$$\bar{x}_d$$ is the mean of the sample differences. $$\mu_{\mathrm{d}}$$ is the mean of the population differences. $$s_d$$ is the sample standard deviation of the differences. $$n$$ is the sample size.

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

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