10.5 Hypothesis Testing for Two Means and Two Proportions

Comparing two population means and proportions is crucial in statistics. We use two-sample tests to determine if there are significant differences between groups, like average heights of men and women or defect rates from different factories.

Interpreting results involves assessing p-values and confidence intervals. Small p-values suggest strong evidence against the null hypothesis, while confidence intervals help estimate the range of plausible differences. It's important to consider both statistical and practical significance when drawing conclusions.

Comparing Two Population Means and Proportions

Two-sample tests for means vs proportions

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• Two-sample t-test for means compares the means of two independent populations (e.g., average heights of men and women)
  • Assumes both populations are normally distributed
  • Test statistic $t$ measures the difference between sample means relative to the variability in the data and sample sizes
    • $\bar{x}_1$ and $\bar{x}_2$ represent the sample means (e.g., average heights in the sample)
    • $μ_1$ and $μ_2$ represent the hypothesized population means (e.g., claimed average heights)
    • $s_1^2$ and $s_2^2$ represent the sample variances (measure of variability in each sample)
    • $n_1$ and $n_2$ represent the sample sizes (number of individuals in each sample)
  • Can be conducted as a two-tailed test or one-tailed test, depending on the research question
• Two-sample z-test for proportions compares the proportions of two independent populations (e.g., proportion of defective products from two factories)
  • Assumes both sample sizes are large enough to approximate a normal distribution
  • Test statistic $z$ measures the difference between sample proportions relative to the pooled sample proportion and sample sizes
    • $\hat{p}_1$ and $\hat{p}_2$ represent the sample proportions (e.g., proportion of defective products in each sample)
    • $p_1$ and $p_2$ represent the hypothesized population proportions (e.g., claimed proportions of defective products)
    • $\hat{p}$ represents the pooled sample proportion (overall proportion of successes in both samples combined)
    • $x_1$ and $x_2$ represent the number of successes in each sample (e.g., number of defective products)

Confidence intervals for population differences

• Confidence interval for the difference between two means (independent samples) estimates the range of plausible values for the true difference in population means
  • Formula combines the difference in sample means with a margin of error based on the t-distribution
    • $t_{\alpha/2}$ represents the critical value from the t-distribution with degrees of freedom based on the smaller sample size minus one
• Confidence interval for the difference between two proportions (independent samples) estimates the range of plausible values for the true difference in population proportions
  • Formula combines the difference in sample proportions with a margin of error based on the standard normal distribution
    • $z_{\alpha/2}$ represents the critical value from the standard normal distribution corresponding to the desired confidence level

Additional Considerations

• Paired samples: When data points in two groups are naturally paired or matched, a paired t-test is used instead of the independent two-sample t-test
• Power analysis: Used to determine the sample size needed to detect a specific effect size with a given level of confidence
• Effect size: Measures the magnitude of the difference between two groups, providing information about the practical significance of the results

Interpreting Results and Drawing Conclusions

Interpretation of statistical results

• P-value interpretation involves assessing the strength of evidence against the null hypothesis
  • A small p-value (typically < 0.05) suggests strong evidence against the null hypothesis, indicating a significant difference between the populations (e.g., there is a significant difference in average heights between men and women)
  • A large p-value (> 0.05) suggests insufficient evidence to reject the null hypothesis, indicating no significant difference between the populations (e.g., there is no significant difference in the proportion of defective products between the two factories)
• Confidence interval interpretation involves assessing whether the interval contains the hypothesized difference
  • If the confidence interval does not contain 0 (for differences in means) or the hypothesized difference (for differences in proportions), it suggests a significant difference between the populations (e.g., the 95% confidence interval for the difference in average heights between men and women does not contain 0, indicating a significant difference)
  • If the confidence interval contains 0 or the hypothesized difference, it suggests no significant difference between the populations (e.g., the 95% confidence interval for the difference in proportions of defective products contains 0, indicating no significant difference)
• Drawing conclusions requires considering both statistical significance and practical significance
  • Statistical significance indicates whether the observed difference is likely due to chance or a real difference in the populations
  • Practical significance considers the magnitude of the difference and its relevance in the specific context (e.g., a statistically significant difference in average heights of 1 cm may not be practically meaningful, while a difference of 10 cm could have important implications)
  • Interpret the results in the context of the original research question and consider potential limitations or confounding factors that could affect the interpretation (e.g., the samples may not be representative of the entire population, or there may be other variables influencing the outcome)