MATLAB: An Introduction with Applications
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Author: Amos Gilat
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![### Understanding Variability in Cross-Country Running Splits
In college, talented runners may join a cross-country team. Runners tend to run their best times when they run even splits. Even splits occur when the runners maintain an even pace throughout the race.
The cross-country coach wants to estimate the typical variability in his best runner's 1-mile splits. He takes a random sample of 25 of this runner's mile splits and finds that this runner's mean 1-mile split is 5.44 minutes per mile, with a standard deviation of 0.14 minutes per mile. This runner's 1-mile splits follow a normal distribution.
#### Task (a):
Find the chi-square critical values \( \chi_L^2 \) and \( \chi_U^2 \) to be used in constructing a 95% confidence interval for the true population standard deviation. (Round your answers to two decimal places.)
\[ \chi_L^2 = \_\_\_\_\_\_ \]
\[ \chi_U^2 = \_\_\_\_\_\_ \]
#### Task (b):
Find the 95% confidence interval for the true variability in his best runner's 1-mile splits. (Round your answers to three decimal places.)
- Lower bound: \_\_\_\_\_\_
- Upper bound: \_\_\_\_\_\_
This problem involves understanding the role of variability and standard deviation in measuring performance consistency. By analyzing the data through chi-square critical values and confidence intervals, we can make informed estimates about true performance variability.](https://content.bartleby.com/qna-images/question/9870bfeb-e26e-42f8-8342-bb7dac6347ce/f1342f13-27fa-4eb3-9f1a-a61a6f9ffe22/c8uyiwu_processed.jpeg)
Transcribed Image Text:### Understanding Variability in Cross-Country Running Splits
In college, talented runners may join a cross-country team. Runners tend to run their best times when they run even splits. Even splits occur when the runners maintain an even pace throughout the race.
The cross-country coach wants to estimate the typical variability in his best runner's 1-mile splits. He takes a random sample of 25 of this runner's mile splits and finds that this runner's mean 1-mile split is 5.44 minutes per mile, with a standard deviation of 0.14 minutes per mile. This runner's 1-mile splits follow a normal distribution.
#### Task (a):
Find the chi-square critical values \( \chi_L^2 \) and \( \chi_U^2 \) to be used in constructing a 95% confidence interval for the true population standard deviation. (Round your answers to two decimal places.)
\[ \chi_L^2 = \_\_\_\_\_\_ \]
\[ \chi_U^2 = \_\_\_\_\_\_ \]
#### Task (b):
Find the 95% confidence interval for the true variability in his best runner's 1-mile splits. (Round your answers to three decimal places.)
- Lower bound: \_\_\_\_\_\_
- Upper bound: \_\_\_\_\_\_
This problem involves understanding the role of variability and standard deviation in measuring performance consistency. By analyzing the data through chi-square critical values and confidence intervals, we can make informed estimates about true performance variability.
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