### 5K Fun Run Time Analysis#### BackgroundThe average time to complete a 5K fun run is 22 minutes, with a standard deviation of 2.6 minutes. We are analyzing the run times of 50 randomly selected runners. Assume a normal distribution and round all answers to 4 decimal places.#### Questions and Answers**a. What is the distribution of $ X $?**$ X \sim N(22, 2.6) $Explanation: - $ X $ denotes the time taken by an individual runner.- Follows a normal distribution with a mean ($ \mu $) of 22 minutes and a standard deviation ($ \sigma $) of 2.6 minutes.**b. What is the distribution of $ \bar{x} $?**$ \bar{x} \sim N(22, 0.3677) $Explanation:- $ \bar{x} $ is the sample mean of the 50 runners.- The standard error of the mean is calculated as $ \frac{\sigma}{\sqrt{n}} = \frac{2.6}{\sqrt{50}} \approx 0.3677 $.**c. What is the distribution of $ \sum x $?**$ \sum x \sim N(1100, 18.384) $Explanation:- $ \sum x $ is the sum of the times of the 50 runners.- The mean of $ \sum x $ is $ n \times \mu = 50 \times 22 = 1100 $.- The standard deviation of $ \sum x $ is $ \sqrt{n} \times \sigma = \sqrt{50} \times 2.6 \approx 18.384 $.**d. If one randomly selected runner is timed, what is the probability that this runner's time will be between 21.9484 and 22.3484 minutes?**\[ P(21.9484 < X < 22.3484) = 0.0612 \]Explanation:- The probability is calculated using the cumulative distribution function (CDF) of the normal distribution given the specified interval.**e. For the 50 runners, find the probability that their average time is between 21.9484 and 22.3484 minutes.**\[ P(21.9484

The average time to run the 5K fun run is 22 minutes and the standard deviation is 2.6 minutes. 50 runners are randomly selected to run the 5K fun run. Round all answers to 4 decimal places where possible and assume a normal distribution. a. What is the distribution of X? X - N( 22 2.6 b. What is the distribution of x? ¤ - N( 22 .3677 v) o Σ c. What is the distribution of ) x? > a x - N( 1100 18.384 v) o o d. If one randomly selected runner is timed, find the probability that this runner's time will be between 21.9484 and 22.3484 minutes. .0612 e. For the 50 runners, find the probability that their average time is between 21.9484 and 22.3484 minutes. .3841 f. Find the probability that the randomly selected 50 person team will have a total time less than 1070. .0514 g. For part e) and f), is the assumption of normal necessary? O NoO Yes o h. The top 20% of all 50 person team relay races will compete in the championship round. These are the 20% lowest times. What is the longest total time that a relay team can have and still make it to the championship round? minutes

The average time to run the 5K fun run is 22 minutes and the standard deviation is 2.6 minutes. 50 runners are randomly selected to run the 5K fun run. Round all answers to 4 decimal places where possible and assume a normal distribution. a. What is the distribution of X? X - N( 22 2.6 b. What is the distribution of x? ¤ - N( 22 .3677 v) o Σ c. What is the distribution of ) x? > a x - N( 1100 18.384 v) o o d. If one randomly selected runner is timed, find the probability that this runner's time will be between 21.9484 and 22.3484 minutes. .0612 e. For the 50 runners, find the probability that their average time is between 21.9484 and 22.3484 minutes. .3841 f. Find the probability that the randomly selected 50 person team will have a total time less than 1070. .0514 g. For part e) and f), is the assumption of normal necessary? O NoO Yes o h. The top 20% of all 50 person team relay races will compete in the championship round. These are the 20% lowest times. What is the longest total time that a relay team can have and still make it to the championship round? minutes

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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