The Boston Marathon has had a wheelchair division since 1977...

The Boston Marathon has had a wheelchair division since $1977 .$ Who do you think is typically faster, the men's marathon winner on foot or the women's wheelchair marathon winner? Because the conditions differ from year to year, and speeds have improved over the years, it seems best to treat these as paired measurements. Here are summary statistics for the pairwise differences in finishing time (in minutes): (www.boston.com/sports/marathon/history/champions/) c) Would a hypothesis test at $\alpha=0.05$ reject the null hypothesis of no difference? What conclusion would you draw?

The Boston Marathon has had a wheelchair division since $1977 .$ Who do you think is typically faster, the men's marathon winner on foot or the women's wheelchair marathon winner? Because the conditions differ from year to year, and speeds have improved over the years, it seems best to treat these as paired measurements. Here are summary statistics for the pairwise differences in finishing time (in minutes):
(www.boston.com/sports/marathon/history/champions/)
c) Would a hypothesis test at $\alpha=0.05$ reject the null hypothesis of no difference? What conclusion would you draw?

Key Concepts

Hypothesis Testing

Hypothesis testing involves setting up a null hypothesis—which in this context states that there is no difference between the two comparisons—and an alternative hypothesis which suggests that a difference exists. The process determines whether the observed differences are statistically significant by calculating a test statistic and corresponding p-value.

Significance Level and Decision Making

The significance level, denoted by ? (in this case, 0.05), is the threshold used to decide whether to reject the null hypothesis. If the p-value computed from the paired t-test is less than ?, the null hypothesis is rejected, indicating a statistically significant difference between the paired groups. This decision process forms the basis for drawing conclusions about the effectiveness or performance differences in the analysis.

Paired Design

In a paired design, observations are taken in pairs that are naturally linked, which allows the analysis to control for variability between the pairs. This design is especially effective when comparing two related methods or treatments, as it provides a more accurate reflection of the differences by accounting for the inherent paired structure in the data.

Paired t-Test

The paired t-test is a statistical method used to compare the means of two related groups by analyzing the differences within each pair. It helps determine whether the average difference between pairs is statistically significantly different from zero, making it ideal for evaluating performance or treatment effects in paired study designs.

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