Let x denote the time it takes to run a road race. Suppose x is approximately normally distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this roa race in less than 177 minutes?
Let x denote the time it takes to run a road race. Suppose x is approximately normally distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this roa race in less than 177 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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![### Probability and Statistics: Normal Distribution
**Problem Statement:**
Let \( x \) denote the time it takes to run a road race. Suppose \( x \) is approximately normally distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this road race in less than 177 minutes?
**Solution Steps:**
1. **Identify the given parameters:**
- Mean (\( \mu \)): 190 minutes
- Standard deviation (\( \sigma \)): 21 minutes
- Target time (\( X = 177 \)) minutes
2. **Standardize the target time using the Z-score formula:**
\[
Z = \frac{X - \mu}{\sigma}
\]
Substitute the given values:
\[
Z = \frac{177 - 190}{21} = \frac{-13}{21} \approx -0.6190
\]
3. **Find the corresponding probability:**
Using standard normal distribution tables, or a calculator, find the probability associated with a Z-score of -0.6190.
**Expected Answer Format:**
Round your answer to four decimal places.
\[
\boxed{}
\]
**Additional Information:**
- The blue icon with an "i" in it typically denotes an information or help button that can provide guidance or clarification when you hover over or click on it.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcce67d60-ba9c-4b65-a73f-671ae19935ed%2F1fce6c11-7526-47e7-9197-f70f7d34a0de%2Fw1rl6je_processed.png&w=3840&q=75)
Transcribed Image Text:### Probability and Statistics: Normal Distribution
**Problem Statement:**
Let \( x \) denote the time it takes to run a road race. Suppose \( x \) is approximately normally distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this road race in less than 177 minutes?
**Solution Steps:**
1. **Identify the given parameters:**
- Mean (\( \mu \)): 190 minutes
- Standard deviation (\( \sigma \)): 21 minutes
- Target time (\( X = 177 \)) minutes
2. **Standardize the target time using the Z-score formula:**
\[
Z = \frac{X - \mu}{\sigma}
\]
Substitute the given values:
\[
Z = \frac{177 - 190}{21} = \frac{-13}{21} \approx -0.6190
\]
3. **Find the corresponding probability:**
Using standard normal distribution tables, or a calculator, find the probability associated with a Z-score of -0.6190.
**Expected Answer Format:**
Round your answer to four decimal places.
\[
\boxed{}
\]
**Additional Information:**
- The blue icon with an "i" in it typically denotes an information or help button that can provide guidance or clarification when you hover over or click on it.
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