A physical trainer has four workouts that he recommends for his clients. The workouts have been designed so that the average maximum heart rate achieved is the same for each workout. To test this design he randomly selects twenty people and randomly assigns five of them to use each of the workouts. During each workout, he measures the maximum heart rate in beats per minute with the following results. Can the physical trainer conclude that there is a difference among the average maximum heart rates which are achieved during the four workouts? Maximum Heart Rates (Beats per Minute)Workout #1Workout #2Workout #3Workout #4

161

189

173

156

157

188

159

158

173

165

169

173

165

185

150

170

168

178

162

164 Find the value of the test statistic to test for a difference between the workouts. Round your answer to two decimal places, if necessary.

To determine whether there is a statistically significant difference in the average maximum heart rates achieved during the four different workouts, we can conduct a **one-way ANOVA (Analysis of Variance)**. This test compares the means across multiple groups (in this case, the four workouts) to determine if at least one of the groups has a significantly different mean.

### Steps for conducting a one-way ANOVA:
1. **Null Hypothesis (\(H_0\))**: The average maximum heart rates are the same across all four workouts.
   \[
   H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4
   \]
   
2. **Alternative Hypothesis (\(H_a\))**: At least one workout has a different average maximum heart rate.
   \[
   H_a: \text{At least one } \mu \text{ is different.}
   \]

3. **Test Statistic**: The ANOVA F-statistic is calculated as:
   \[
   F = \frac{\text{Mean Square Between (MSB)}}{\text{Mean Square Within (MSW)}}
   \]

4. **Degrees of Freedom**:
   - Between groups: \( df_{between} = k - 1 \), where \( k \) is the number of groups (workouts).
   - Within groups: \( df_{within} = N - k \), where \( N \) is the total number of observations.

### Data Summary:
- Number of groups (\( k \)) = 4 (four workouts)
- Total number of observations (\( N \)) = 20

The data for each workout is:

- Workout #1: \( 161, 157, 173, 169, 170 \)
- Workout #2: \( 189, 188, 173, 173, 168 \)
- Workout #3: \( 173, 159, 165, 178, 162 \)
- Workout #4: \( 156, 158, 165, 150, 164 \)

I'll now calculate the ANOVA F-statistic.The value of the test statistic (F-statistic) for the one-way ANOVA is **5.49**.

This test statistic can now be compared to a critical value from the F-distribution or used to calculate the p-value to determine if there is a statistically significant difference among the average maximum heart rates for the four workouts.

Would you like to see more details on how to interpret the result or calculate the p-value?

Here are five related questions to explore further:
1. What is the p-value corresponding to this F-statistic?
2. How do you interpret the F-statistic in ANOVA tests?
3. What is the critical F-value for a specific significance level (e.g., 0.05) in this test?
4. What assumptions must be met to properly conduct a one-way ANOVA?
5. How can post-hoc tests (like Tukey’s test) help identify which workouts differ?

**Tip**: ANOVA tests check for differences between group means, but they don't tell you which specific groups differ. For that, you need to conduct post-hoc tests.

A physical trainer has four workouts that he recommends for his clients. The workouts have been designed so that the average maximum heart rate achieved is the same for each workout. To test this design he randomly selects twenty people and randomly assigns five of them to use each of the workouts. During each workout, he measures the maximum heart rate in beats per minute with the following results. Can the physical trainer conclude that there is a difference among the average maximum heart rates which are achieved during the four workouts? Maximum Heart Rates (Beats per Minute)Workout #1: 161, 157, 173, 169, 170Workout #2: 189, 188, 173, 173, 168Workout #3: 173, 159, 165, 178, 162Workout #4: 156, 158, 165, 150, 164. Find the value of the test statistic to test for a difference between the workouts. Round your answer to two decimal places, if necessary.

Learn how to conduct a one-way ANOVA to determine if there is a difference in the average maximum heart rates achieved during four workouts. This statistical analysis includes the calculation of the F-statistic and interpretation of the test result. Suitable for college-level statistics students.

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