### Calculating the Area of a Shaded Region in a Normal DistributionTo determine the area under the normal distribution curve, follow these steps:#### Problem StatementFind 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.](#)#### Graph DescriptionThe graph shows a normal distribution curve with a mean of 100 and a standard deviation of 15. The shaded region under the curve spans from an IQ score of 80 to 125.#### Calculation RequiredThe area of the shaded region is: ____ (Round to four decimal places as needed)By clicking the provided table links, you can use the standard normal distribution table (Z-table) to find the respective Z-scores for the interval [80, 125]. Summarize your findings, and then fill in the blank with the area calculated, rounded to four decimal places. ## Understanding Normal Distribution of IQ Scores### Problem Statement:Assume that adults have IQ scores that are normally distributed with a mean of 100.8 and a standard deviation of 16.3. Find the probability that a randomly selected adult has an IQ greater than 121.2. (Hint: Draw a graph.)### Calculation:The probability that a randomly selected adult from this group has an IQ greater than 121.2 is _____. (Round to four decimal places as needed.)### Explanation:To solve this problem, you need to calculate the Z-score for 121.2 using the given mean and standard deviation. You can then use the Z-score to find the probability from the standard normal distribution table.1. **Calculate the Z-score:** \[ Z = \frac{X - \mu}{\sigma} \] Where: - $ X $ is the value for which you want to find the probability (121.2), - $ \mu $ is the mean (100.8), - $ \sigma $ is the standard deviation (16.3).2. **Find the Probability:** Once you have the Z-score, use the standard normal distribution table or a graphing tool to find the probability that the IQ is greater than 121.2.3. **Graphical Representation:** Draw a bell curve representing the normal distribution of IQ scores with the center at 100.8. Shade the area to the right of 121.2. This shaded area represents the probability that an adult's IQ is greater than 121.2.This problem helps in understanding how to use normal distribution to find probabilities of a specific range of data.

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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.

MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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Concept explainers

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

Question

### Calculating the Area of a Shaded Region in a Normal Distribution

To determine the area under the normal distribution curve, follow these steps:

#### 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.](#)

#### Graph Description
The graph shows a normal distribution curve with a mean of 100 and a standard deviation of 15. The shaded region under the curve spans from an IQ score of 80 to 125.

#### Calculation Required
The area of the shaded region is: ____ (Round to four decimal places as needed)

By clicking the provided table links, you can use the standard normal distribution table (Z-table) to find the respective Z-scores for the interval [80, 125]. Summarize your findings, and then fill in the blank with the area calculated, rounded to four decimal places.

## Understanding Normal Distribution of IQ Scores

### Problem Statement:

Assume that adults have IQ scores that are normally distributed with a mean of 100.8 and a standard deviation of 16.3. Find the probability that a randomly selected adult has an IQ greater than 121.2. (Hint: Draw a graph.)

### Calculation:

The probability that a randomly selected adult from this group has an IQ greater than 121.2 is _____. (Round to four decimal places as needed.)

### Explanation:

To solve this problem, you need to calculate the Z-score for 121.2 using the given mean and standard deviation. You can then use the Z-score to find the probability from the standard normal distribution table.

1. **Calculate the Z-score:**
\[
Z = \frac{X - \mu}{\sigma}
\]
Where:
- $ X $ is the value for which you want to find the probability (121.2),
- $ \mu $ is the mean (100.8),
- $ \sigma $ is the standard deviation (16.3).

2. **Find the Probability:**
Once you have the Z-score, use the standard normal distribution table or a graphing tool to find the probability that the IQ is greater than 121.2.

3. **Graphical Representation:**
Draw a bell curve representing the normal distribution of IQ scores with the center at 100.8. Shade the area to the right of 121.2. This shaded area represents the probability that an adult's IQ is greater than 121.2.

This problem helps in understanding how to use normal distribution to find probabilities of a specific range of data.

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