Two-Sample Tests and One-Way ANOVA: Comparing Means, Proportions, and Variances

Two-Sample Tests and One-Way ANOVA

Introduction

This chapter covers statistical methods for comparing the means, proportions, and variances of two or more populations. These methods are essential for business decision-making, allowing analysts to determine if observed differences are statistically significant or due to random variation.

Two-Sample Tests

Overview of Two-Sample Tests

• Population Means, Independent Samples: Compare means from two unrelated groups (e.g., Group 1 vs. Group 2).

• Population Means, Related Samples: Compare means from the same group before and after treatment (paired or matched samples).

• Population Proportions: Compare proportions from two groups (e.g., Proportion 1 vs. Proportion 2).

• Population Variances: Compare variances from two groups (e.g., Variance 1 vs. Variance 2).

Difference Between Two Means

Independent Samples

To test hypotheses or form confidence intervals for the difference between two population means ($\mu_1 - \mu_2$), use:

• Pooled-Variance t Test: When population variances are unknown but assumed equal.

• Separate-Variance t Test: When population variances are unknown and not assumed equal.

The point estimate for the difference is $\overline{X}_1 - \overline{X}_2$.

Assumptions for Independent Samples

• Samples are randomly and independently drawn.

• Populations are normally distributed or both sample sizes are at least 30.

• For pooled-variance t test: Population variances are assumed equal.

• For separate-variance t test: Population variances are not assumed equal.

Hypothesis Tests for Two Population Means

• Lower-tail test: $H_0: \mu_1 - \mu_2 \geq 0$ vs. $H_1: \mu_1 - \mu_2 < 0$

• Upper-tail test: $H_0: \mu_1 - \mu_2 \leq 0$ vs. $H_1: \mu_1 - \mu_2 > 0$

• Two-tail test: $H_0: \mu_1 - \mu_2 = 0$ vs. $H_1: \mu_1 - \mu_2 \neq 0$

Reject $H_0$ if the test statistic falls in the critical region determined by the significance level $\alpha$.

Pooled-Variance t Test

• Pooled Variance: $S_p^2 = \frac{(n_1 - 1)S_1^2 + (n_2 - 1)S_2^2}{n_1 + n_2 - 2}$

• Test Statistic: $t_{STAT} = \frac{(\overline{X}_1 - \overline{X}_2) - (\mu_1 - \mu_2)}{\sqrt{S_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

• Degrees of freedom: $n_1 + n_2 - 2$

Confidence Interval for $\mu_1 - \mu_2$ (Pooled Variance)

• $\left(\overline{X}_1 - \overline{X}_2\right) \pm t_{\alpha/2} \sqrt{S_p^2 \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}$

• Where $t_{\alpha/2}$ is the critical value from the t-distribution with $n_1 + n_2 - 2$ degrees of freedom.

Example: Pooled-Variance t Test

Sales Location	Sample Mean ($\overline{X}$)	Sample Variance ($S^2$)	n
Special Front	246.4	42.5420	10
In-Aisle	202.3	32.5271	10

• Test statistic: $t = 2.6041$

• Critical value at $\alpha = 0.05$: $t_{0.025,18} = 2.1009$

• Decision: Reject $H_0$; there is evidence of a difference in means.

Separate-Variance t Test

• Used when population variances are unknown and not assumed equal.

• Test statistic and degrees of freedom are calculated using software.

• Example: Comparing dividend yields between NYSE and NASDAQ stocks.

Related Populations: The Paired Difference Test

Paired Samples

• Used for matched or paired samples (e.g., before/after measurements).

• Eliminates variation among subjects by focusing on differences within pairs.

• Assumptions: Differences are normally distributed or sample size is large.

Test Statistic for Paired Difference

• Let $D_i = X_{1i} - X_{2i}$ be the difference for pair $i$.

• Sample mean of differences: $\overline{D} = \frac{\sum D_i}{n}$

• Sample standard deviation: $S_D = \sqrt{\frac{\sum (D_i - \overline{D})^2}{n-1}}$

• Test statistic: $t_{STAT} = \frac{\overline{D} - \mu_D}{S_D / \sqrt{n}}$

• Degrees of freedom: $n-1$

Confidence Interval for Paired Difference

• $\overline{D} \pm t_{\alpha/2} \frac{S_D}{\sqrt{n}}$

Example: Paired Difference Test

Item	Costco	Walmart	Difference
Chicken Broth	5.98	5.88	0.10
Ice Cream	8.59	7.19	1.40
Dishwasher Detergent	9.00	17.00	-8.00
Laundry Detergent	11.00	12.00	-1.00
Paper Towels	1.47	2.09	-0.62
Toilet Paper	12.00	27.00	-15.00
Facial Tissues	1.23	1.12	0.11

• $\overline{D} = -3.2781$

• $S_D = 6.0101$

Two Population Proportions

Testing the Difference Between Proportions

• Goal: Test hypothesis or form a confidence interval for $p_1 - p_2$.

• Assumptions: $n_1p_1, n_1(1-p_1), n_2p_2, n_2(1-p_2) \geq 5$

• Pooled estimate for overall proportion: $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$

• Test statistic: $Z_{STAT} = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

Hypothesis Tests for Two Proportions

• Lower-tail test: $H_0: p_1 - p_2 \geq 0$ vs. $H_1: p_1 - p_2 < 0$

• Upper-tail test: $H_0: p_1 - p_2 \leq 0$ vs. $H_1: p_1 - p_2 > 0$

• Two-tail test: $H_0: p_1 - p_2 = 0$ vs. $H_1: p_1 - p_2 \neq 0$

Confidence Interval for Two Proportions

• $(\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2} \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

Comparing Two Population Variances

F Test for Equality of Variances

• Hypotheses: $H_0: \sigma_1^2 = \sigma_2^2$ vs. $H_1: \sigma_1^2 \neq \sigma_2^2$

• Test statistic: $F = \frac{S_1^2}{S_2^2}$ (larger variance in numerator)

• Degrees of freedom: $df_1 = n_1 - 1$, $df_2 = n_2 - 1$

• Compare calculated $F$ to critical value from F-distribution table.

One-Way Analysis of Variance (ANOVA)

Purpose and Design

• Used to compare means of three or more groups.

• Assumptions: Populations are normally distributed, have equal variances, and samples are randomly and independently selected.

• Completely randomized design: Subjects are randomly assigned to groups.

Hypotheses for One-Way ANOVA

• Null hypothesis ($H_0$): All population means are equal ($\mu_1 = \mu_2 = \ldots = \mu_c$).

• Alternative hypothesis ($H_1$): At least one population mean is different.

Partitioning the Variation

• Total Sum of Squares (SST): Total variation among all data values.

• Sum of Squares Among Groups (SSA): Variation among group means.

• Sum of Squares Within Groups (SSW): Variation within each group.

• Relationship: $SST = SSA + SSW$

Formulas

• $SST = \sum_{j=1}^c \sum_{i=1}^{n_j} (X_{ij} - \overline{X})^2$

• $SSA = \sum_{j=1}^c n_j (\overline{X}_j - \overline{X})^2$

• $SSW = \sum_{j=1}^c \sum_{i=1}^{n_j} (X_{ij} - \overline{X}_j)^2$

Mean Squares and F Statistic

• Mean Square Among: $MSA = \frac{SSA}{c-1}$

• Mean Square Within: $MSW = \frac{SSW}{n-c}$

• F Statistic: $F = \frac{MSA}{MSW}$

• Degrees of freedom: $df_1 = c-1$, $df_2 = n-c$

Interpreting the F Statistic

• If $F$ is greater than the critical value from the F-distribution, reject $H_0$.

• Conclusion: At least one group mean is different.

Assumptions for ANOVA

• Randomness and independence of samples.

• Normality of populations.

• Homogeneity of variances (can be tested with Levene's Test).

When Assumptions Are Violated

• If only normality is violated: Use Kruskal-Wallis rank test.

• If only equal variance is violated: Use separate-variance procedures.

• If both are violated: Data transformation is needed.

Levene's Test for Homogeneity of Variance

• Tests whether group variances are equal.

• Null hypothesis: All group variances are equal.

• Procedure: Compute absolute deviations from group medians and perform ANOVA on these values.

Post-Hoc Comparisons: Tukey-Kramer Procedure

• Used after a significant ANOVA F test to determine which means differ.

• Compares absolute mean differences to a critical range based on the studentized range distribution.

Summary Table: One-Way ANOVA

Source	Sum of Squares	Degrees of Freedom	Mean Square	F
Among Groups	SSA	c-1	MSA	MSA/MSW
Within Groups	SSW	n-c	MSW
Total	SST	n-1

Chapter Summary

• Compared means and proportions of two independent populations.

• Compared means of two related populations.

• Compared variances of two independent populations.

• Compared means and variances of more than two populations using ANOVA.