For problem 3, assume the random variable x is normally distributed with a mean of 86 and a standard deviation of 5. Find the indicated probability. 3. P(x > 89)
For problem 3, assume the random variable x is normally distributed with a mean of 86 and a standard deviation of 5. Find the indicated probability. 3. P(x > 89)
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 Problem
**Problem Statement:**
For problem 3, assume the random variable \( x \) is normally distributed with a mean of 86 and a standard deviation of 5. Find the indicated probability.
**Problem 3:**
\[ \text{P}(x > 89) \]
### Explanation:
In this problem, we are given a normally distributed random variable \( x \) with a specified mean (μ) and standard deviation (σ).
- **Mean (μ)**: 86
- **Standard Deviation (σ)**: 5
We need to find the probability that \( x \) takes on a value greater than 89, denoted as \( \text{P}(x > 89) \).
To solve this, we'll use the properties of the normal distribution:
1. **Standardize the Variable**: Convert \( x \) to a standard normal variable \( z \) using the formula:
\[ z = \frac{x - \mu}{\sigma} \]
2. **Calculate the Z-Score**: For \( x = 89 \),
\[ z = \frac{89 - 86}{5} = \frac{3}{5} = 0.6 \]
3. **Find the Probability**: Look up the Z-score in the standard normal distribution table (or use a Z-score calculator) to find the probability that \( z \) is less than 0.6. Subtract this value from 1 to find \( \text{P}(z > 0.6) \).
Remember,
\[ \text{P}(x > 89) = \text{P}(z > 0.6) \]
### Conclusion:
Using the standard normal distribution table or a calculator will give you the numerical result for the probability \( \text{P}(x > 89) \). This completes the solution approach for the given problem on normal distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F685c5768-905e-4931-8257-6730855b6d77%2Fc3ac5fab-9e2a-41ed-9a2b-d627460bd3a8%2Fc6zz85n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Normal Distribution Problem
**Problem Statement:**
For problem 3, assume the random variable \( x \) is normally distributed with a mean of 86 and a standard deviation of 5. Find the indicated probability.
**Problem 3:**
\[ \text{P}(x > 89) \]
### Explanation:
In this problem, we are given a normally distributed random variable \( x \) with a specified mean (μ) and standard deviation (σ).
- **Mean (μ)**: 86
- **Standard Deviation (σ)**: 5
We need to find the probability that \( x \) takes on a value greater than 89, denoted as \( \text{P}(x > 89) \).
To solve this, we'll use the properties of the normal distribution:
1. **Standardize the Variable**: Convert \( x \) to a standard normal variable \( z \) using the formula:
\[ z = \frac{x - \mu}{\sigma} \]
2. **Calculate the Z-Score**: For \( x = 89 \),
\[ z = \frac{89 - 86}{5} = \frac{3}{5} = 0.6 \]
3. **Find the Probability**: Look up the Z-score in the standard normal distribution table (or use a Z-score calculator) to find the probability that \( z \) is less than 0.6. Subtract this value from 1 to find \( \text{P}(z > 0.6) \).
Remember,
\[ \text{P}(x > 89) = \text{P}(z > 0.6) \]
### Conclusion:
Using the standard normal distribution table or a calculator will give you the numerical result for the probability \( \text{P}(x > 89) \). This completes the solution approach for the given problem on normal distribution.
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