The area of the shaded region is
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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![**Calculating the Area of the Shaded Region in a Normal Distribution**
**Problem Statement:**
Find the area of the shaded region. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.
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**Graph Explanation:**
A bell-shaped normal distribution curve is depicted, representing the distribution of IQ scores. The mean (μ) of the IQ scores is 100, and the standard deviation (σ) is 15. A vertical line at the IQ score of 95 marks the shaded region on the left side of this line under the curve.
**Objective:**
Calculate the area of the shaded region to represent the probability that an individual's IQ score is less than 95.
**Solution:**
The area of the shaded region is the probability \( P(X < 95) \) under the normal curve, which can be calculated using the Z-score formula:
\[ Z = \frac{X - \mu}{\sigma} \]
By substituting the given values:
- X = 95
- μ = 100
- σ = 15
\[ Z = \frac{95 - 100}{15} = \frac{-5}{15} = - \frac{1}{3} \approx -0.3333 \]
Using the Z-table, we find the cumulative probability associated with \( Z = -0.3333 \).
**To be Determined:**
The area of the shaded region is \(\boxed{(\text{rounded observation from Z-table})}\). (Round to four decimal places as needed.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadca8183-f536-4768-a5dc-a2639f5a41ee%2F2864b8a5-38c0-46fd-8806-5607155b6636%2Fghym17b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculating the Area of the Shaded Region in a Normal Distribution**
**Problem Statement:**
Find the area of the shaded region. The graph to the right depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.
[Click to view page 1 of the table.] [Click to view page 2 of the table.]
(Insert Table Links)
**Graph Explanation:**
A bell-shaped normal distribution curve is depicted, representing the distribution of IQ scores. The mean (μ) of the IQ scores is 100, and the standard deviation (σ) is 15. A vertical line at the IQ score of 95 marks the shaded region on the left side of this line under the curve.
**Objective:**
Calculate the area of the shaded region to represent the probability that an individual's IQ score is less than 95.
**Solution:**
The area of the shaded region is the probability \( P(X < 95) \) under the normal curve, which can be calculated using the Z-score formula:
\[ Z = \frac{X - \mu}{\sigma} \]
By substituting the given values:
- X = 95
- μ = 100
- σ = 15
\[ Z = \frac{95 - 100}{15} = \frac{-5}{15} = - \frac{1}{3} \approx -0.3333 \]
Using the Z-table, we find the cumulative probability associated with \( Z = -0.3333 \).
**To be Determined:**
The area of the shaded region is \(\boxed{(\text{rounded observation from Z-table})}\). (Round to four decimal places as needed.)
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