This chapter uses the example of comparing breakfast habits east and west of the Mississippi River to introduce hypothesis testing across two groups. It covers classifying hypothesis tests by type, testing two population means with both known and unknown standard deviations, testing two population proportions, and handling matched or paired samples.
Classifying two-sample hypothesis tests
The chapter opens by teaching students to classify hypothesis tests by type, since comparing two groups requires first identifying whether the groups are independent or matched, and whether the comparison concerns means or proportions, before the correct testing procedure can be chosen.
Testing two population means
Students learn to conduct and interpret hypothesis tests for two population means both when the population standard deviations are known and when they are unknown, mirroring the one-sample case from the previous chapter but now comparing two groups directly, such as breakfast habits in different regions.
Testing proportions and matched pairs
The chapter also covers conducting and interpreting hypothesis tests for two population proportions and for matched or paired samples, where the same subjects are measured under two different conditions, requiring a method that accounts for the dependence between the paired observations rather than treating them as two independent groups.
This chapter extends hypothesis testing to comparisons between two groups, using business examples such as night-shift versus day-shift productivity and the investment returns of two different strategies. It distinguishes independent groups from matched pairs and covers testing two population means and two population proportions under each design.
Independent groups versus matched pairs
Comparing two groups requires first classifying them as independent, where sample values from one population are unrelated to those from the other, or as matched pairs, where the two samples are dependent, and the chapter uses this distinction to determine which testing method and which parameter, means or proportions, applies.
Comparing two independent population means
The comparison of two independent population means is presented as a common business question, illustrated with whether the night shift is less productive than the day shift or whether one investment strategy’s returns differ from another’s, framing the two-sample mean comparison as a natural extension of the one-sample methods covered earlier.
Comparing two proportions and paired samples
Beyond comparing means, the chapter covers hypothesis tests for two population proportions, useful for comparing rates such as production efficiency under different management styles, and for matched or paired samples, where the same subjects or units are measured under two conditions and the pairing itself must be accounted for in the test.
Many questions compare two groups rather than one. This chapter extends hypothesis testing to two samples, showing how to test the difference between two population means when standard deviations are known or unknown, how to compare two population proportions, and how to handle matched or paired samples where observations come in pairs.
Comparing Two Population Means
When two groups are compared, the hypotheses concern the difference between their population means. If the population standard deviations are known, the normal distribution is used; if they are unknown and estimated from the samples, the Student t distribution is used instead. The test statistic measures how far the observed difference between the two sample means lies from the value stated in the null hypothesis, usually a difference of zero, relative to the variability of the data.
Comparing Two Proportions
To compare two population proportions, such as the success rates of two treatments, the difference between the two sample proportions is examined. Under the assumption that the proportions are equal, the sampling distribution of their difference is approximately normal for large samples, so a z-based test applies. As with means, the observed difference is standardised and turned into a p-value that measures how surprising the result would be if the two proportions were truly equal.
Matched or Paired Samples
Sometimes the two samples are not independent but naturally paired, such as the same subjects measured before and after a treatment. In this case the analysis works with the differences within each pair rather than the two groups separately. The set of paired differences is then treated as a single sample, and a one-sample test on those differences checks whether the average change differs from zero, controlling for variation between individuals.
