Heights of men on a baseball team have a​ bell-shaped distribution with a mean of 173 cm and a standard deviation of 9 cm. Using the empirical​ rule, what is the approximate percentage of the men between the following​ values?
a. 146 cm and 200 cm
b. 155 cm and 191 cm
Question content area bottom
Part 1
a. 
  
enter your response here​% of the men are between 146 cm and 200 cm.

The empirical rule (or 68-95-99.7 rule) tells us how data is distributed in a normal (bell-shaped) distribution in terms of standard deviations from the mean:

- **68%** of data falls within 1 standard deviation of the mean.
- **95%** falls within 2 standard deviations.
- **99.7%** falls within 3 standard deviations.

Given that the mean is 173 cm and the standard deviation is 9 cm, let's calculate for the two cases:

### Part a: Between 146 cm and 200 cm

First, calculate how many standard deviations these values are away from the mean:
- \( 146 \, \text{cm} \) is: 
  \[
  \frac{173 - 146}{9} = 3 \, \text{standard deviations below the mean.}
  \]
- \( 200 \, \text{cm} \) is: 
  \[
  \frac{200 - 173}{9} = 3 \, \text{standard deviations above the mean.}
  \]

According to the empirical rule, approximately **99.7%** of the data falls within 3 standard deviations of the mean.

Therefore, **99.7%** of the men are between 146 cm and 200 cm.

### Part b: Between 155 cm and 191 cm

Similarly, calculate how many standard deviations these values are away from the mean:
- \( 155 \, \text{cm} \) is: 
  \[
  \frac{173 - 155}{9} = 2 \, \text{standard deviations below the mean.}
  \]
- \( 191 \, \text{cm} \) is: 
  \[
  \frac{191 - 173}{9} = 2 \, \text{standard deviations above the mean.}
  \]

According to the empirical rule, approximately **95%** of the data falls within 2 standard deviations of the mean.

Therefore, **95%** of the men are between 155 cm and 191 cm.

---

Let me know if you'd like more details or have any questions.

### Follow-up questions:
1. What does the empirical rule say about data falling within 1 standard deviation of the mean?
2. How do we use the z-score to find probabilities in a normal distribution?
3. What is the importance of the empirical rule in statistics?
4. Can you explain why the empirical rule works only for normal distributions?
5. How would you calculate percentages if the distribution wasn’t normal?

**Tip**: Always check whether the distribution is approximately normal before applying the empirical rule.

Heights of men on a baseball team have a bell-shaped distribution with a mean of 173 cm and a standard deviation of 9 cm. Using the empirical rule, what is the approximate percentage of the men between the following values? a. 146 cm and 200 cm b. 155 cm and 191 cm

Learn how to use the empirical rule to calculate the percentage of men's heights on a baseball team that fall between specific values. This problem demonstrates the use of the empirical rule and standard deviation for normally distributed data, with a mean of 173 cm and a standard deviation of 9 cm. Suitable for 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