Two-Sample Tests: Comparing Means and Variances in Business Statistics

Two-Sample Tests

Introduction

Two-sample tests are essential statistical tools used to compare the means or variances of two populations. In business statistics, these tests help determine whether differences observed between groups (such as sales at two locations or prices at two stores) are statistically significant. This chapter covers hypothesis testing and confidence interval estimation for two population means and variances, using both independent and related samples.

Difference Between Two Means

Independent Samples

When comparing two population means using independent samples, the goal is to test hypotheses or construct confidence intervals for the difference between the means ($\mu_1 - \mu_2$).

• Independent samples: Data from one population does not affect the other.

• Point estimate for the difference: $\bar{X}_1 - \bar{X}_2$

• Variance assumptions:

  • Unknown but assumed equal: Use pooled-variance t test.

  • Unknown and not assumed equal: Use separate-variance t test.

Hypothesis Tests for Two Population Means

Types of Tests

Depending on the research question, hypothesis tests can be lower-tail, upper-tail, or two-tail.

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

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

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

Rejection regions are determined by the critical values of the t-distribution:

• Lower-tail: Reject $H_0$ if $t_{STAT} < -t_{\alpha}$

• Upper-tail: Reject $H_0$ if $t_{STAT} > t_{\alpha}$

• Two-tail: Reject $H_0$ if $t_{STAT} < -t_{\alpha/2}$ or $t_{STAT} > t_{\alpha/2}$

Pooled-Variance t Test

Assumptions

• Samples are randomly and independently drawn.

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

• Population variances are unknown but assumed equal.

Formulas

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

• Degrees of freedom: $df = n_1 + n_2 - 2$

Confidence Interval for $\mu_1 - \mu_2$

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

• Where $t_{\alpha/2}$ is based on $df = n_1 + n_2 - 2$

Pooled-Variance t Test Example

Suppose you compare sales at two store locations:

• Special Front: $\bar{X}_1 = 246.4$, $S_1^2 = 42.5420$, $n_1 = 10$

• In-Aisle: $\bar{X}_2 = 202.3$, $S_2^2 = 32.5271$, $n_2 = 10$

Calculate pooled variance and test statistic:

• $S_p^2 = 1433.9167$

• $t_{STAT} = 2.6041$

• Critical value for $\alpha = 0.05$ (two-tailed): $t_{0.025,18} = 2.1009$

• Decision: Since $2.6041 > 2.1009$, reject $H_0$; there is evidence of a difference in means.

Excel Output and P-value

• P-value = 0.0179 < 0.05, so $H_0$ is rejected.

• Interpretation: The probability of observing such a difference by chance is 0.0179.

Summary Table

Result	Conclusions
$t_{STAT} = 2.6041 > 2.1009$	Reject $H_0$. Mean sales are different for the two locations.
P-value = 0.0179 < 0.05	Probability of observing such a difference is 0.0179.
$t_{STAT}$ is positive	Mean sales are higher for the special front location.

Evaluating the Normality Assumption

• Pooled-variance t test assumes normality and equal variances.

• Use box plots (e.g., in Excel) to visually check normality for each group.

Related Populations: The Paired Difference Test

Paired Samples

Used when samples are related (e.g., before/after measurements, matched pairs). This test eliminates variation among subjects by focusing on the difference within each pair.

• Paired difference: $D_i = X_{1i} - X_{2i}$

• Point estimate for mean difference: $\bar{D} = \frac{\sum D_i}{n}$

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

• Assumptions: Differences are normally distributed, or use large samples.

Test Statistic for Paired Difference

• $t_{STAT} = \frac{\bar{D} - \mu_D}{S_D / \sqrt{n}}$

• Degrees of freedom: $n - 1$

Possible Hypotheses

• Lower-tail: $H_0: \mu_D \geq 0$ vs. $H_1: \mu_D < 0$

• Upper-tail: $H_0: \mu_D \leq 0$ vs. $H_1: \mu_D > 0$

• Two-tail: $H_0: \mu_D = 0$ vs. $H_1: \mu_D \neq 0$

Confidence Interval for Paired Difference

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

Paired Difference Test Example

A researcher compares the mean price of a market basket at Costco and Walmart for $n = 7$ items:

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	27.00	-15.00
Toilet Paper	12.00	12.00	0.00
Facial Tissues	1.23	1.12	0.11

• $\bar{D} = -3.2781$

• $S_D = 6.0101$

• $t_{STAT} = -1.4471$

• Critical value $t_{0.025,6} = 2.4469$

• Decision: Do not reject $H_0$; insufficient evidence of a difference in mean price.

The F Distribution

Comparing Variances

The F distribution is used to compare the variances of two populations. The F critical value is found from the F table, requiring two degrees of freedom: numerator and denominator.

• Test statistic: $F_{STAT} = \frac{S_1^2}{S_2^2}$

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

• Numerator: Larger sample variance

• Rejection region:

  • Two-tailed: Reject $H_0$ if $F_{STAT} > F_{\alpha/2}$

  • One-tailed: Reject $H_0$ if $F_{STAT} > F_{\alpha}$

Summary Table: Hypothesis Test Results

Result	Conclusions
$t_{STAT} = 2.6041 > 2.1009$	Reject $H_0$. Mean sales are different for the two locations.
P-value = 0.0179 < 0.05	Probability of observing such a difference is 0.0179.
$t_{STAT}$ is positive	Mean sales are higher for the special front location.
$t_{STAT} = -1.4471 < 2.4469$	Do not reject $H_0$. No evidence of a difference in mean price between Costco and Walmart.
P-value = 0.1980 > 0.05	No evidence exists that there is a difference in the mean price of equivalent items purchased at Costco and Walmart.

Key Takeaways

• Use two-sample t tests to compare means from independent or related samples.

• Check assumptions of normality and equal variances before applying pooled-variance t tests.

• Use the F test to compare variances between two populations.

• Excel can be used to perform these tests and visualize data distributions.