https://file.aiquickdraw.com/mathbot/file/1729487696363e3zug4yk.jpg

The problem you shared asks to find the area of the shaded region under a normal distribution curve representing IQ scores. The distribution is normally distributed with a mean (\(\mu\)) of 100 and a standard deviation (\(\sigma\)) of 15. The shaded area corresponds to IQ scores less than 95.

### Steps to solve:

1. **Standardize the score**:
   To find the area under the normal curve, we first need to convert the IQ score of 95 to a z-score using the formula:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   where:
   - \(X = 95\) (the IQ score in question),
   - \(\mu = 100\) (mean),
   - \(\sigma = 15\) (standard deviation).

   Substituting the values:
   \[
   z = \frac{95 - 100}{15} = \frac{-5}{15} = -0.3333
   \]

2. **Find the cumulative probability**:
   Using the z-score of -0.3333, we now refer to the standard normal distribution table or use a calculator to find the cumulative probability.

   The cumulative probability corresponding to \(z = -0.3333\) is approximately 0.3707.

3. **Interpretation**:
   This means the area under the normal curve to the left of \(z = -0.3333\), which corresponds to IQ scores less than 95, is about 0.3707 or 37.07%.

Thus, the area of the shaded region is **0.3707** (rounded to four decimal places).

Would you like further details on this or any clarifications?

### Related Questions:
1. How do you find the area between two z-scores on a normal distribution?
2. What does the area under a normal distribution curve represent?
3. How do we use the standard normal distribution table?
4. What is the empirical rule in a normal distribution?
5. How do different standard deviations affect the shape of a normal distribution curve?

### Tip:
When solving normal distribution problems, always standardize the variable first by converting it into a z-score, then use standard normal tables or a calculator for probabilities.

Find the area of the shaded region. The graph depicts IQ scores of adults, which are normally distributed with a mean of 100 and a standard deviation of 15. The shaded region corresponds to IQ scores less than 95.

This page explains how to find the area of the shaded region under a normal distribution curve with a mean of 100 and standard deviation of 15, corresponding to IQ scores below 95. The solution uses z-scores and cumulative probability.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation