Find the area between.x = 18 and .x = 23 under a normal distribution curve with μ = 20 and o = 4. Round your answer to four decimal places.
Find the area between.x = 18 and .x = 23 under a normal distribution curve with μ = 20 and o = 4. Round your answer to four decimal places.
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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![### Problem Statement
**Objective:** Find the area between \( x = 18 \) and \( x = 23 \) under a normal distribution curve with \( \mu = 20 \) and \( \sigma = 4 \).
**Instructions:** Round your answer to four decimal places.
### Explanation and Solution
To find the area under a normal distribution curve between two points, you need to calculate the cumulative probabilities for each point and then determine the difference between these probabilities.
1. **Standardize the values of \( x \):**
Standardized values (z-scores) can be computed as:
\[
z = \frac{x - \mu}{\sigma}
\]
For \( x = 18 \):
\[
z_1 = \frac{18 - 20}{4} = \frac{-2}{4} = -0.5
\]
For \( x = 23 \):
\[
z_2 = \frac{23 - 20}{4} = \frac{3}{4} = 0.75
\]
2. **Use the z-scores to find cumulative probabilities** from the standard normal distribution table or a computational tool:
\[
P(Z < -0.5) \approx 0.3085
\]
\[
P(Z < 0.75) \approx 0.7734
\]
3. **Calculate the area between these two points** by finding the difference between the cumulative probabilities:
\[
\text{Area} = P(Z < 0.75) - P(Z < -0.5)
\]
\[
\text{Area} = 0.7734 - 0.3085 = 0.4649
\]
Therefore, the area between \( x = 18 \) and \( x = 23 \) under the specified normal distribution curve is approximately \( 0.4649 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcce67d60-ba9c-4b65-a73f-671ae19935ed%2F119ca195-cc4b-4f25-8fc8-2b3835658f57%2Ff8hncm_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Objective:** Find the area between \( x = 18 \) and \( x = 23 \) under a normal distribution curve with \( \mu = 20 \) and \( \sigma = 4 \).
**Instructions:** Round your answer to four decimal places.
### Explanation and Solution
To find the area under a normal distribution curve between two points, you need to calculate the cumulative probabilities for each point and then determine the difference between these probabilities.
1. **Standardize the values of \( x \):**
Standardized values (z-scores) can be computed as:
\[
z = \frac{x - \mu}{\sigma}
\]
For \( x = 18 \):
\[
z_1 = \frac{18 - 20}{4} = \frac{-2}{4} = -0.5
\]
For \( x = 23 \):
\[
z_2 = \frac{23 - 20}{4} = \frac{3}{4} = 0.75
\]
2. **Use the z-scores to find cumulative probabilities** from the standard normal distribution table or a computational tool:
\[
P(Z < -0.5) \approx 0.3085
\]
\[
P(Z < 0.75) \approx 0.7734
\]
3. **Calculate the area between these two points** by finding the difference between the cumulative probabilities:
\[
\text{Area} = P(Z < 0.75) - P(Z < -0.5)
\]
\[
\text{Area} = 0.7734 - 0.3085 = 0.4649
\]
Therefore, the area between \( x = 18 \) and \( x = 23 \) under the specified normal distribution curve is approximately \( 0.4649 \).
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