Assume the random variable x is normally distributed with mean μ = 80 and standard deviation o=4. Find the indicated probability. P(68
Assume the random variable x is normally distributed with mean μ = 80 and standard deviation o=4. Find the indicated probability. P(68
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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![**Understanding Normal Distribution Probability Calculation**
In this example, we are given a normally distributed random variable \( x \) with a mean (\( \mu \)) of 80 and a standard deviation (\( \sigma \)) of 4. The task is to find the probability that \( x \) lies between 68 and 77, expressed as \( P(68 < x < 77) \).
**Steps to Solve:**
1. **Identify the Parameters of the Normal Distribution:**
- Mean (\( \mu \)) = 80
- Standard deviation (\( \sigma \)) = 4
2. **Standardize the Variable:**
- Convert the raw scores into z-scores using the formula:
\[
z = \frac{x - \mu}{\sigma}
\]
3. **Calculate the z-scores:**
- For \( x = 68 \):
\[
z = \frac{68 - 80}{4} = -3
\]
- For \( x = 77 \):
\[
z = \frac{77 - 80}{4} = -0.75
\]
4. **Use the Standard Normal Distribution Table:**
- Find the probability for each z-score.
- The probability for \( z = -3 \) and \( z = -0.75 \) can be found from a standard normal distribution table or a calculator that provides cumulative probabilities.
5. **Compute the Probability:**
- The desired probability \( P(68 < x < 77) \) is the difference between the cumulative probabilities:
\[
P(-3 < z < -0.75) = P(z < -0.75) - P(z < -3)
\]
6. **Round to Four Decimal Places:**
- Ensure final answers are rounded to four decimal places for precision.
This process involves understanding basic concepts of normal distribution, standard deviation, and how cumulative probability tables work. Solving this will provide a clearer understanding of how probability is calculated within specified intervals in a normal distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26146d8a-c15c-4983-bbd8-fc4fc59b6ca5%2F3d4ee02c-0e7b-474b-a53b-94554e6952f5%2Fypx1uyq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Normal Distribution Probability Calculation**
In this example, we are given a normally distributed random variable \( x \) with a mean (\( \mu \)) of 80 and a standard deviation (\( \sigma \)) of 4. The task is to find the probability that \( x \) lies between 68 and 77, expressed as \( P(68 < x < 77) \).
**Steps to Solve:**
1. **Identify the Parameters of the Normal Distribution:**
- Mean (\( \mu \)) = 80
- Standard deviation (\( \sigma \)) = 4
2. **Standardize the Variable:**
- Convert the raw scores into z-scores using the formula:
\[
z = \frac{x - \mu}{\sigma}
\]
3. **Calculate the z-scores:**
- For \( x = 68 \):
\[
z = \frac{68 - 80}{4} = -3
\]
- For \( x = 77 \):
\[
z = \frac{77 - 80}{4} = -0.75
\]
4. **Use the Standard Normal Distribution Table:**
- Find the probability for each z-score.
- The probability for \( z = -3 \) and \( z = -0.75 \) can be found from a standard normal distribution table or a calculator that provides cumulative probabilities.
5. **Compute the Probability:**
- The desired probability \( P(68 < x < 77) \) is the difference between the cumulative probabilities:
\[
P(-3 < z < -0.75) = P(z < -0.75) - P(z < -3)
\]
6. **Round to Four Decimal Places:**
- Ensure final answers are rounded to four decimal places for precision.
This process involves understanding basic concepts of normal distribution, standard deviation, and how cumulative probability tables work. Solving this will provide a clearer understanding of how probability is calculated within specified intervals in a normal distribution.
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