91Ó°ÊÓ

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. $$ \begin{array}{ccc|ccc} \text { Woman } & \text { Before } & \text { After } & \text { Woman } & \text { Before } & \text { After } \\ \hline 1 & 23.5 & 19.75 & 6 & 19.75 & 19.5 \\ \hline 2 & 18.5 & 19.25 & 7 & 35 & 34.25 \\ \hline 3 & 21.5 & 21.75 & 8 & 36.5 & 35 \\ \hline 4 & 24 & 22.5 & 9 & 52 & 51.5 \\ \hline 5 & 25 & 25 & 10 & 30 & 31 \\ \hline \end{array} $$

Step by step solution

01

- State the Null and Alternative Hypotheses

The null hypothesis (H_0) assumes that the median waistline before the exercise program is equal to the median waistline after the exercise program. The alternative hypothesis (H_a) assumes that the median waistline before the exercise program is more than the median waistline after the exercise program.

02

- Organize the Data

List the waistline measurements before and after the exercise program: Before: [23.5, 18.5, 21.5, 24, 25, 19.75, 35, 36.5, 52, 30] After: [19.75, 19.25, 21.75, 22.5, 25, 19.5, 34.25, 35, 51.5, 31]

03

- Calculate the Medians

Order the data and find the middle value for both sets. Ordered Before: [18.5, 19.75, 21.5, 23.5, 24, 25, 30, 35, 36.5, 52] Ordered After: [19.25, 19.5, 19.75, 21.75, 22.5, 25, 31, 34.25, 35, 51.5] Median Before: (24 + 25) / 2 = 24.5 Median After: (22.5 + 25) / 2 = 23.75

04

- Perform the Sign Test

Determine the differences between the pairs and count the number of times the after measurement is smaller than the before measurement. Differences: [-3.75, 0.75, -0.25, 1.5, 0, -0.25, 0.75, 1.5, 0.5, -1] Negative differences: 4 out of 10 times

05

- Determine the Critical Value

Using the binomial distribution table for α = 0.05 with n = 10, find the critical value. Since this is a one-tailed test, the critical value is the number of negative differences at P < 0.05.

06

- Make a Decision

Compare the number of negative differences to the critical value. Since 4 is less than the critical value found in the binomial distribution table, reject the null hypothesis.

07

- Conclusion

Conclude that there is significant evidence at the α = 0.05 level to say that the median waistline before the exercise program is more than the median waistline after the exercise program.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. It involves forming assumptions (called hypotheses) and then using statistical techniques to determine whether these assumptions hold true. In our example, the physical therapist wants to test whether the median waistline before the exercise program is more than the median waistline after the exercise program, with a significance level (\(\alpha\=0.05\)) of 0.05.
The steps to hypothesis testing generally include:

• Formulating the null hypothesis (H0) and the alternative hypothesis (Ha)
• Choosing a significance level (\(\alpha\))
• Collecting data and calculating the test statistic
• Comparing the test statistic to a critical value
• Making a decision to either reject or fail to reject the null hypothesis

In our context, the null hypothesis (H0) assumes no effect (median waistlines are equal), and the alternative hypothesis (Ha) assumes the exercise program is effective (median waistline before is more than after).

Sign Test

The Sign Test is a non-parametric test used to determine if there is a significant difference between the medians of two related groups. It is especially useful when the sample size is small or when the data does not satisfy the assumptions required for parametric tests, like the normal distribution.
In our exercise, the Sign Test assesses whether the median waistline before the exercise program is more than that after the exercise program. Here's how it's done:

• Calculate the differences between the pairs of measurements (Before - After)
• Classify the differences as positive, negative, or zero (differences of zero are usually ignored)
• Count the number of positive and negative differences
• Use the binomial distribution table to find the critical value for the given significance level (\(\alpha\=0.05\))
• Compare the count of negative differences to the critical value to make a decision

In our case, we found 4 negative differences out of 10 paired measurements. Since 4 is less than the critical value found in the binomial distribution table, we reject the null hypothesis.

Median Comparison

Comparing medians is a common way to understand the central tendency of data, especially when dealing with non-normally distributed data. The median is the middle value of a data set when ordered from least to greatest, providing a robust measure of central location.
In our exercise:

• The data before the exercise program, ordered: [18.5, 19.75, 21.5, 23.5, 24, 25, 30, 35, 36.5, 52]
• The data after the exercise program, ordered: [19.25, 19.5, 19.75, 21.75, 22.5, 25, 31, 34.25, 35, 51.5]

The median is found by averaging the two middle values in each set since there are an even number of observations:

• Median Before: (\(\frac{24 + 25}\2\)) = 24.5
• Median After: (\(\frac{22.5 + 25}\2\)) = 23.75

The comparison shows that the median waistline was indeed greater before the exercise program. This supports the therapist's assumption, leading to a plausible conclusion when combined with the statistical testing.