Comparing Means and Proportions: Two-Sample Tests and ANOVA

Chapter 10: Two-Sample Tests and Analysis of Variance (ANOVA)

Comparing Means of Two Populations

Statistical methods allow us to compare the means of two populations to determine if there is a significant difference. This is commonly done using hypothesis testing.

• Hypothesis Setup: Formulate null and alternative hypotheses based on the research question.

  • Two-tailed: $H_0: \mu_1 = \mu_2$ vs. $H_a: \mu_1 \neq \mu_2$

  • Lower-tailed: $H_0: \mu_1 \geq \mu_2$ vs. $H_a: \mu_1 < \mu_2$

  • Upper-tailed: $H_0: \mu_1 \leq \mu_2$ vs. $H_a: \mu_1 > \mu_2$

• Types of t-Tests: Depending on the data and assumptions, different t-tests are used:

  • Paired Two Sample for Means: Used when samples are related (e.g., before-and-after measurements).

  • Two-Sample Assuming Equal Variances: Used when both populations are assumed to have the same variance.

  • Two-Sample Assuming Unequal Variances: Used when population variances are not assumed equal.

• Excel Implementation: Excel provides built-in functions for these t-tests, such as T.TEST and Data Analysis Toolpak.

• Interpretation: After performing the test, interpret the p-value:

  • If p-value < significance level (e.g., 0.05), reject the null hypothesis.

  • If p-value > significance level, fail to reject the null hypothesis.

• Example: Comparing the average sales between two regions using a two-sample t-test.

Comparing Proportions of Two Populations

When the variable of interest is categorical, we may compare proportions between two populations.

• Hypothesis Setup:

  • Null Hypothesis: $H_0: p_1 = p_2$

  • Alternative Hypothesis: $H_a: p_1 \neq p_2$ (two-tailed), $p_1 < p_2$ (lower-tailed), or $p_1 > p_2$ (upper-tailed)

• Test Statistic: The test statistic for comparing proportions is: $z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$ where $\hat{p}$ is the pooled proportion.

• Excel Implementation: Use formulas or Data Analysis Toolpak for proportion tests.

• Interpretation: Similar to t-tests, interpret the p-value to determine statistical significance.

• Example: Comparing the proportion of customers who prefer product A in two different markets.

Comparing Means of Two or More Populations: ANOVA

When comparing means across more than two groups, Analysis of Variance (ANOVA) is used to test if at least one group mean is different.

• Hypothesis Setup:

  • Null Hypothesis: $H_0: \mu_1 = \mu_2 = \ldots = \mu_k$

  • Alternative Hypothesis: At least one mean differs.

• ANOVA Test Statistic: The F-statistic is calculated as: $F = \frac{\text{Between-group variance}}{\text{Within-group variance}}$

• Excel Implementation: Use ANOVA: Single Factor in Data Analysis Toolpak.

• Interpretation: If the p-value is less than the significance level, reject the null hypothesis and conclude that at least one mean is different.

• Example: Comparing average sales across three different regions using ANOVA.

Summary Table: Types of Tests for Comparing Groups

Test Type	Purpose	Assumptions	Excel Tool
Paired t-Test	Compare means of related samples	Samples are paired/matched	T.TEST, Data Analysis Toolpak
Two-Sample t-Test (Equal Variances)	Compare means of two independent samples	Equal population variances	T.TEST, Data Analysis Toolpak
Two-Sample t-Test (Unequal Variances)	Compare means of two independent samples	Unequal population variances	T.TEST, Data Analysis Toolpak
Proportion Test	Compare proportions of two groups	Binomial distribution, large sample	Formulas, Data Analysis Toolpak
ANOVA	Compare means of three or more groups	Normality, equal variances	ANOVA: Single Factor

Additional info: The notes reference Excel implementation, which is commonly used in business statistics courses for hypothesis testing and ANOVA.