Effects of Exercise To determine the effectiveness of an...

Effects of Exercise To determine the effectiveness of an exercise regimen, a physical therapist randomly selects 10 women to participate in a study. She measures their waistlines (in inches) both before and after a rigorous 8 -week exercise program and obtains the data shown. Is the median waistline before the exercise program more than the median waistline after the exercise program? Use the $\alpha=0.05$ level of significance. $$ \begin{array}{ccc|ccc} \text { Woman } & \text { Before } & \text { After } & \text { Woman } & \text { Before } & \text { After } \\ \hline 1 & 23.5 & 19.75 & 6 & 19.75 & 19.5 \\ \hline 2 & 18.5 & 19.25 & 7 & 35 & 34.25 \\ \hline 3 & 21.5 & 21.75 & 8 & 36.5 & 35 \\ \hline 4 & 24 & 22.5 & 9 & 52 & 51.5 \\ \hline 5 & 25 & 25 & 10 & 30 & 31 \\ \hline \end{array} $$

Effects of Exercise To determine the effectiveness of an exercise regimen, a physical therapist randomly selects 10 women to participate in a study. She measures their waistlines (in inches) both before and after a rigorous 8 -week exercise program and obtains the data shown. Is the median waistline before the exercise program more than the median waistline after the exercise program? Use the $\alpha=0.05$ level of significance.
$$
\begin{array}{ccc|ccc}
\text { Woman } & \text { Before } & \text { After } & \text { Woman } & \text { Before } & \text { After } \\
\hline 1 & 23.5 & 19.75 & 6 & 19.75 & 19.5 \\
\hline 2 & 18.5 & 19.25 & 7 & 35 & 34.25 \\
\hline 3 & 21.5 & 21.75 & 8 & 36.5 & 35 \\
\hline 4 & 24 & 22.5 & 9 & 52 & 51.5 \\
\hline 5 & 25 & 25 & 10 & 30 & 31 \\
\hline
\end{array}
$$

Key Concepts

Hypothesis Testing

This concept involves formulating a null hypothesis (which, in this context, typically states that there is no difference in median waistlines before and after the exercise program) and an alternative hypothesis (suggesting that the median waistline before the program is greater than after). The process requires evaluating the evidence from the data against a predetermined significance level to decide whether to reject the null hypothesis.

Paired Sample Design

In a paired sample design, measurements are taken on the same subjects under different conditions—in this case, before and after the exercise regimen. This approach controls for individual variability since each participant serves as her own control, allowing for a more direct comparison of the two sets of measurements.

Significance Level

The significance level, denoted by ?, is the threshold for statistical decision-making. With ? = 0.05, there is a 5% risk of rejecting the null hypothesis when it is actually true. This level of significance is used to determine whether the observed differences in median waistlines are likely due to the effect of the exercise regimen or merely due to chance.

Nonparametric Methods (Wilcoxon Signed-Rank Test)

When the focus is on median differences rather than mean differences, or when data do not meet the assumptions required for parametric tests, nonparametric tests such as the Wilcoxon signed-rank test are employed. This test is appropriate for paired samples and assesses whether there is a systematic difference in the median values between the two related groups.

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