Comparing Two Means

Introduction

In statistics, comparing the means of two groups is a fundamental method for determining whether a significant difference exists between them. This is commonly encountered in experiments and observational studies, such as evaluating the effect of an incentive on customer spending or comparing outcomes before and after an intervention.

Two-Sample t-Methods

Difference of Means

The primary statistic of interest is the difference between the means of two independent groups. In practice, we estimate this difference using sample means, but our goal is to infer about the population means.

• Population parameter of interest:

• Sample statistic:

Sampling Distribution and Standard Error

To determine if the observed difference is statistically significant, we need the sampling distribution and standard error of the difference between means. If the samples are independent, the variance of the difference is the sum of the variances.

• Standard deviation of the difference:

• Estimated standard error (when population standard deviations are unknown):

Student's t Distribution

The ratio of the observed difference in means to its standard error follows a Student's t distribution with adjusted degrees of freedom. This allows us to perform hypothesis tests and construct confidence intervals.

Two-Sample t-Test

The two-sample t-test is used to test hypotheses about the difference between two means. The most common null hypothesis is that the means are equal ().

• Test statistic:

Assumptions and Conditions

• Independence Assumption: Data in each group must be drawn independently and at random.

• Randomization Condition: Data should be collected with suitable randomization.

• 10% Condition: If sampling without replacement, the sample should be no more than 10% of the population.

• Normal Population Assumption: Each group should come from a population that is approximately Normal.

• Independent Groups Assumption: The two groups must be independent of each other.

Confidence Interval for the Difference Between Two Means

A confidence interval provides a range of plausible values for the true difference in population means. The interval is constructed as follows:

$

Example: Website Design

A market analyst compares the mean number of pages viewed on two different website designs. The data are summarized below:

Hypothesis test results: P-value = 0.5573. Fail to reject the null hypothesis; there is no evidence of a difference in mean page visits. The 95% confidence interval (–0.938, 1.738) includes 0, supporting the same conclusion.

Pooled t-Test

When to Use the Pooled t-Test

The pooled t-test is used when the variances of the two groups are assumed to be equal. This method pools the sample variances to obtain a common estimate, which can increase the test's power when the assumption is justified.

• Equal Variance Assumption:

Pooled Standard Error

Pooled t-Test and Confidence Interval

The test statistic and confidence interval are similar to the two-sample t-test, but use the pooled standard error and degrees of freedom .

Advice: Because the advantages of pooling are small and the equal variances assumption is rarely justified, it is generally safer not to pool.

Example: Buying a Used Camera

Paired Data and Paired t-Methods

Paired Data

Paired data arise when the same subjects are measured under two conditions, such as before and after an intervention. The analysis focuses on the differences within each pair.

• Paired t-test: A one-sample t-test applied to the differences between paired observations.

Assumptions for Paired t-Test

• Paired Data Assumption: Data must be paired (e.g., before and after measurements on the same subjects).

• Independence Assumption: The differences must be independent of each other.

• Randomization Condition: Treatments should be randomly assigned.

• 10% Condition: If sampling without replacement, the sample should be no more than 10% of the population.

• Normal Population Assumption: The population of differences should be approximately Normal.

Paired t-Test Formula

The paired t-test statistic is:

• is the mean of the pairwise differences

Paired t Confidence Interval

The confidence interval for the mean paired difference is:

Example: ERP System Implementation

• Paired data assumption: Data are paired by company.

• Randomization condition: The 8 companies are assumed to be representative.

• Normal population assumption: A histogram of differences is needed to check Normality.

• Result: 95% CI for mean difference (0.053, 1.847), t = 2.50, P-value = 0.041. At , reject the null hypothesis; there is evidence that the average acceleration time is different after implementation.

Summary of Key Points

• The two-sample t-test is appropriate for comparing means of independent groups.

• Assumptions include independence, randomization, and Normality.

• Pooled t-tests require equal variances; use with caution.

• Paired t-tests are used for matched or paired samples, focusing on differences within pairs.

• Always check assumptions and conditions before applying these methods.