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there is a 5k race, the recorded time was normally distributed with a mean time of 28 and a standard deviation of 5 minutes. it is a known that 20% of the

there is a 5k race, the recorded time was normally distributed with a mean time of 28 and a standard deviation of 5 minutes. it is a known that 20% of the

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.3: Special Probability Density Functions
Problem 7E
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### Understanding Normal Distribution in 5K Race Times In a study of a 5K race, the time taken to complete the race was found to be normally distributed. The key statistical measures are: - **Mean Time:** 28 minutes - **Standard Deviation:** 5 minutes From the data, it is known that: - 20% of the runners took more than 28 minutes but less than \( k \) minutes to complete the race. ### Problem Statement We need to determine the value of \( k \) using the principles of normal distribution. ### Solution Approach 1. **Identify the Mean and Standard Deviation:** - Mean (\( \mu \)) = 28 minutes - Standard Deviation (\( \sigma \)) = 5 minutes 2. **Understand the Normal Distribution:** - The normal distribution is symmetric about the mean. - The area under the curve represents the probability. 3. **Calculate the Z-Score:** - The Z-score represents the number of standard deviations a data point is from the mean. - \( Z = \frac{X - \mu}{\sigma} \) 4. **Use the Given Probabilities:** - 20% of the runners took more than 28 minutes but less than \( k \) minutes. This implies that the area under the normal curve from a Z-score of 0 to the Z-score corresponding to \( k \) is 0.20. 5. **Find the Corresponding Z-Score:** - Using a Z-table or normal distribution calculator, find the Z-score for which 20% of the distribution lies between 0 and Z. This Z-score corresponds to the upper 20% of the mean. - Let Z be the Z-score such that \( P(0 < Z < z) = 0.20 \). 6. **Calculate \( k \):** - \( k = \mu + Z \sigma \) - Substitute \( \mu \), \( \sigma \), and the Z value obtained from the table. 7. **Interpret the Value:** - \( k \) will be the time in minutes. By applying these steps, you should be able to determine the value of \( k \) according to the normal distribution laws.
Transcribed Image Text:### Understanding Normal Distribution in 5K Race Times In a study of a 5K race, the time taken to complete the race was found to be normally distributed. The key statistical measures are: - **Mean Time:** 28 minutes - **Standard Deviation:** 5 minutes From the data, it is known that: - 20% of the runners took more than 28 minutes but less than \( k \) minutes to complete the race. ### Problem Statement We need to determine the value of \( k \) using the principles of normal distribution. ### Solution Approach 1. **Identify the Mean and Standard Deviation:** - Mean (\( \mu \)) = 28 minutes - Standard Deviation (\( \sigma \)) = 5 minutes 2. **Understand the Normal Distribution:** - The normal distribution is symmetric about the mean. - The area under the curve represents the probability. 3. **Calculate the Z-Score:** - The Z-score represents the number of standard deviations a data point is from the mean. - \( Z = \frac{X - \mu}{\sigma} \) 4. **Use the Given Probabilities:** - 20% of the runners took more than 28 minutes but less than \( k \) minutes. This implies that the area under the normal curve from a Z-score of 0 to the Z-score corresponding to \( k \) is 0.20. 5. **Find the Corresponding Z-Score:** - Using a Z-table or normal distribution calculator, find the Z-score for which 20% of the distribution lies between 0 and Z. This Z-score corresponds to the upper 20% of the mean. - Let Z be the Z-score such that \( P(0 < Z < z) = 0.20 \). 6. **Calculate \( k \):** - \( k = \mu + Z \sigma \) - Substitute \( \mu \), \( \sigma \), and the Z value obtained from the table. 7. **Interpret the Value:** - \( k \) will be the time in minutes. By applying these steps, you should be able to determine the value of \( k \) according to the normal distribution laws.
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