Math Problem Statement
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.
Solution
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:
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Null Hypothesis (): The average maximum heart rates are the same across all four workouts.
-
Alternative Hypothesis (): At least one workout has a different average maximum heart rate.
-
Test Statistic: The ANOVA F-statistic is calculated as:
-
Degrees of Freedom:
- Between groups: , where is the number of groups (workouts).
- Within groups: , where is the total number of observations.
Data Summary:
- Number of groups () = 4 (four workouts)
- Total number of observations () = 20
The data for each workout is:
- Workout #1:
- Workout #2:
- Workout #3:
- Workout #4:
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:
- What is the p-value corresponding to this F-statistic?
- How do you interpret the F-statistic in ANOVA tests?
- What is the critical F-value for a specific significance level (e.g., 0.05) in this test?
- What assumptions must be met to properly conduct a one-way ANOVA?
- 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.
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Math Problem Analysis
Mathematical Concepts
Statistics
ANOVA
Hypothesis Testing
Formulas
F = MSB / MSW
Degrees of freedom (Between): df_between = k - 1
Degrees of freedom (Within): df_within = N - k
Theorems
One-way ANOVA
Suitable Grade Level
College Level (Statistics)
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