https://file.aiquickdraw.com/mathbot/file/17293032491907c2f4xt6.jpg

The image presents a set of exam scores that are approximately normally distributed with a mean of 63 and a standard deviation of 8. The problem is asking to use the **Empirical Rule** to determine the area (percentage) of the shaded regions in the bell curve for each case.

### Empirical Rule (68-95-99.7 Rule):
The rule states that for a normal distribution:
1. 68% of the data falls within **1 standard deviation** from the mean.
2. 95% of the data falls within **2 standard deviations** from the mean.
3. 99.7% of the data falls within **3 standard deviations** from the mean.

Let's analyze each case:

#### 1st Graph:
- The region shaded is between 55 and 63.
- Since 63 is the mean, 55 is **1 standard deviation below** the mean (63 - 8 = 55).
- According to the Empirical Rule, 34% of the data falls between the mean and 1 standard deviation below (or above).
  
Thus, the percentage for this shaded area is **34%**, which is the correct answer provided in the image.

#### 2nd Graph:
- The region shaded is between 63 and 71.
- 71 is **1 standard deviation above** the mean (63 + 8 = 71).
- As before, 34% of the data lies between the mean and 1 standard deviation above the mean.
  
So, the percentage for this shaded area is also **34%**, which is again correct in the image.

#### 3rd Graph:
- The region shaded is between 55 and 71.
- This covers the area from **1 standard deviation below** the mean (55) to **1 standard deviation above** the mean (71).
- According to the Empirical Rule, 68% of the data falls between 1 standard deviation below and 1 standard deviation above the mean.

So, the percentage for this shaded area is **68%**, but the image shows an incorrect value of 47.5%.

### Final Values:
1. 34% (correct)
2. 34% (correct)
3. 68% (should be corrected from 47.5%)

Would you like more details on the Empirical Rule or any other related concepts? 

Here are five additional questions for further exploration:
1. How is the Empirical Rule derived from the properties of a normal distribution?
2. Can you apply the Empirical Rule to distributions that are not normal?
3. How would the shaded areas change if the standard deviation was 10 instead of 8?
4. What is the significance of the 95% region in real-world applications?
5. How does the Empirical Rule relate to standard normal (Z) scores?

Tip: In normal distributions, standard deviation defines the spread of the data, so knowing it allows for quick estimates of where most data points fall.

A set of exam scores are approximately normally distributed with a mean of 63 and standard deviation of 8. Use the Empirical Rule to determine the area of the shaded regions shown below.

Learn how to apply the Empirical Rule (68-95-99.7 Rule) to a normal distribution with a mean of 63 and a standard deviation of 8. Solve questions involving the area of shaded regions under the bell curve, suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation