A normal distribution has a mean of 32 and a standard deviation of 4. Find the probability that a randomly selected x-value from the distribution is in the given interval + 16 16 + 20 24 28 20 24 28 32 X 32 X % % 36 + 40 44 36 40 44
A normal distribution has a mean of 32 and a standard deviation of 4. Find the probability that a randomly selected x-value from the distribution is in the given interval + 16 16 + 20 24 28 20 24 28 32 X 32 X % % 36 + 40 44 36 40 44
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 Probabilities in a Normal Distribution**
In this educational lesson, we will explore how to find the probability that a randomly selected x-value from a normal distribution falls within a specified interval. Consider the following problem:
---
A normal distribution has a mean \( \mu \) of 32 and a standard deviation \( \sigma \) of 4. We want to find the probability that a randomly selected x-value from the distribution falls within a given interval.
---
### Graphical Explanation:
1. **First Interval: 28 to 36**
In the first graph, we see a normal distribution curve with a shaded blue area representing the interval from 28 to 36.
- The x-axis is labeled with values: 16, 20, 24, 28, 32, 36, 40, 44.
- The shaded region lies between 28 and 36.
To find the probability that a randomly selected x-value falls within this interval:
- Calculate the z-scores for x = 28 and x = 36.
- Find the cumulative probabilities for these z-scores using standard normal distribution tables or software.
- Subtract the cumulative probability at x = 28 from the cumulative probability at x = 36.
Make sure to express the result as a percentage.
\[
\text{Probability (P)} = \left[ \text{P}(x \leq 36) - \text{P}(x \leq 28) \right] \times 100
\]
Enter the result in the box provided:
\[ \_\_\_\_\_\_ \% \]
2. **Second Interval: 24 to 32**
In the second graph, we see another normal distribution curve with the shaded blue area representing the interval from 24 to 32.
- The x-axis is labeled with values: 16, 20, 24, 28, 32, 36, 40, 44.
- The shaded region lies between 24 and 32.
To find the probability that a randomly selected x-value falls within this interval:
- Calculate the z-scores for x = 24 and x = 32.
- Find the cumulative probabilities for these z-scores using standard normal distribution tables or software.
- Subtract the cumulative probability at x = 24 from the cumulative probability at x = 32.
Make sure to express the result as a percentage](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa81afcb-79d6-433d-9647-f14beb19ad20%2F313a87ea-ee47-4c3f-9a19-6de56b191a1c%2Feqcfgxg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Probabilities in a Normal Distribution**
In this educational lesson, we will explore how to find the probability that a randomly selected x-value from a normal distribution falls within a specified interval. Consider the following problem:
---
A normal distribution has a mean \( \mu \) of 32 and a standard deviation \( \sigma \) of 4. We want to find the probability that a randomly selected x-value from the distribution falls within a given interval.
---
### Graphical Explanation:
1. **First Interval: 28 to 36**
In the first graph, we see a normal distribution curve with a shaded blue area representing the interval from 28 to 36.
- The x-axis is labeled with values: 16, 20, 24, 28, 32, 36, 40, 44.
- The shaded region lies between 28 and 36.
To find the probability that a randomly selected x-value falls within this interval:
- Calculate the z-scores for x = 28 and x = 36.
- Find the cumulative probabilities for these z-scores using standard normal distribution tables or software.
- Subtract the cumulative probability at x = 28 from the cumulative probability at x = 36.
Make sure to express the result as a percentage.
\[
\text{Probability (P)} = \left[ \text{P}(x \leq 36) - \text{P}(x \leq 28) \right] \times 100
\]
Enter the result in the box provided:
\[ \_\_\_\_\_\_ \% \]
2. **Second Interval: 24 to 32**
In the second graph, we see another normal distribution curve with the shaded blue area representing the interval from 24 to 32.
- The x-axis is labeled with values: 16, 20, 24, 28, 32, 36, 40, 44.
- The shaded region lies between 24 and 32.
To find the probability that a randomly selected x-value falls within this interval:
- Calculate the z-scores for x = 24 and x = 32.
- Find the cumulative probabilities for these z-scores using standard normal distribution tables or software.
- Subtract the cumulative probability at x = 24 from the cumulative probability at x = 32.
Make sure to express the result as a percentage
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