https://file.aiquickdraw.com/mathbot/file/17265835016439egln9uf.jpg

This problem requires the use of the **empirical rule**, which is applicable for bell-shaped (normal) distributions.

### Key Information:
- **Mean (μ)** = 58
- **Standard deviation (σ)** = 5
- We are asked to find the approximate percentage of daily phone calls between **43 and 73**.

### Steps:
1. **Find the Z-scores**:
   - Convert the values 43 and 73 into Z-scores:
     - \( Z = \frac{X - \mu}{\sigma} \)
   - For 43:
     \[
     Z_1 = \frac{43 - 58}{5} = \frac{-15}{5} = -3
     \]
   - For 73:
     \[
     Z_2 = \frac{73 - 58}{5} = \frac{15}{5} = 3
     \]

2. **Use the empirical rule**:
   - The empirical rule states that:
     - 68% of the data falls within 1 standard deviation from the mean.
     - 95% of the data falls within 2 standard deviations from the mean.
     - 99.7% of the data falls within 3 standard deviations from the mean.
   
   Since 43 and 73 are **3 standard deviations** away from the mean (on both sides), this range covers **99.7%** of the distribution.

Thus, the approximate percentage of daily phone calls between 43 and 73 is **99.7%**.

Would you like more details or have any further questions?

Here are 5 related questions:
1. What is the empirical rule and why is it important in statistics?
2. How do Z-scores help in understanding the normal distribution?
3. What percentage of data lies within 1 standard deviation in a normal distribution?
4. How would the result change if the range was between 48 and 68?
5. How can we use Z-scores to compare data from different distributions?

**Tip**: The empirical rule is most useful when dealing with normal distributions. It provides a quick estimate for data ranges without needing complex calculations.

In a mid-size company, the distribution of the number of phone calls answered each day by each of the 12 receptionists is bell-shaped and has a mean of 58 and a standard deviation of 5. Using the empirical rule, what is the approximate percentage of daily phone calls numbering between 43 and 73?

Solve a statistics problem involving the empirical rule and normal distribution. The problem covers the calculation of the percentage of phone calls between 43 and 73, using a mean of 58 and standard deviation of 5. 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