6.3 Two-sample tests for means and proportions

Two-sample tests help us compare means or proportions between groups. They're crucial for determining if differences are statistically significant, whether we're looking at independent samples or paired data.

These tests have specific assumptions and processes. We'll explore t-tests for means, z-tests for proportions, and how to determine appropriate sample sizes. Understanding these concepts is key for making informed decisions based on data comparisons.

Two-Sample Tests for Means

Two-sample t-test for means

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• Compares means of two independent populations to determine significant differences
• Assumptions: independent samples, approximately normal distributions, equal or unequal variances
• Process:

1. State null and alternative hypotheses
2. Choose significance level (α)
3. Calculate test statistic
4. Determine degrees of freedom
5. Find critical value or p-value
6. Make decision and interpret results

• Test statistic: $t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
• Interpretation: reject or fail to reject null hypothesis, consider practical significance vs statistical significance (SAT scores, drug efficacy)

Paired t-test for dependent samples

• Compares means of dependent samples used for before-after studies or matched pairs (weight loss program, educational intervention)
• Assumptions: dependent samples, differences between pairs normally distributed
• Process:

1. Calculate differences between paired observations
2. Compute mean and standard deviation of differences
3. Calculate test statistic
4. Determine degrees of freedom
5. Find critical value or p-value
6. Make decision and interpret results

• Test statistic: $t = \frac{\bar{d}}{s_d / \sqrt{n}}$
• Interpretation: reject or fail to reject null hypothesis, consider effect size and practical implications

Two-Sample Tests for Proportions

Two-sample z-test for proportions

• Compares proportions of two independent populations to determine significant differences
• Assumptions: independent samples, large sample sizes (np and nq > 5 for both samples)
• Process:

1. State null and alternative hypotheses
2. Choose significance level (α)
3. Calculate pooled proportion
4. Compute test statistic
5. Find critical value or p-value
6. Make decision and draw conclusions

• Test statistic: $z = \frac{(\hat{p}_1 - \hat{p}_2) - (p_1 - p_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$
• Drawing conclusions: interpret statistical significance, consider practical implications (voting patterns, marketing campaign effectiveness)

Sample sizes in two-sample tests

• Factors: desired precision (margin of error), power (1 - β), effect size, significance level (α)
• Two-sample t-test sample size: $n = \frac{2(z_{\alpha/2} + z_{\beta})^2\sigma^2}{\Delta^2}$
  • $\Delta$ represents minimum detectable difference
  • $\sigma^2$ is population variance
• Two-sample test of proportions sample size: $n = \frac{(z_{\alpha/2} + z_{\beta})^2[p_1(1-p_1) + p_2(1-p_2)]}{(p_1 - p_2)^2}$
• Power analysis: examines relationship between sample size and power, balances precision, power, and cost
• Paired designs generally require smaller sample sizes than independent designs, account for correlation between paired observations (clinical trials, psychological studies)