In a recent survey of 655 working Americans
ages 25-34, the average weekly amount spent on lunch was $44.6 with standard deviation $2.81. The weekly amounts are approximatly bell-shaped.
a. Estimate the percentage of amounts that are between $36.17 and $53.03.

To estimate the percentage of lunch spending amounts between \$36.17 and \$53.03, we can apply the **Empirical Rule** (or 68-95-99.7 Rule), which is used for bell-shaped (normally distributed) data.

### Step-by-step process:

1. **Calculate the mean ($ \mu $) and standard deviation ($ \sigma $)**:
   - Mean: $ \mu = 44.6 $
   - Standard deviation: $ \sigma = 2.81 $

2. **Convert the values of \$36.17 and \$53.03 to z-scores** (standardized scores). The formula for the z-score is:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- $x$ is the value
- $ \mu $ is the mean
- $ \sigma $ is the standard deviation

- For \$36.17:
\[
z_1 = \frac{36.17 - 44.6}{2.81} \approx -3.0
\]
- For \$53.03:
\[
z_2 = \frac{53.03 - 44.6}{2.81} \approx 3.0
\]

3. **Interpret using the Empirical Rule**:
   - A z-score of ±3 means that approximately **99.7%** of the data lies within three standard deviations of the mean.

Since the range from \$36.17 to \$53.03 corresponds to $ \mu \pm 3\sigma $, we can estimate that about **99.7%** of the weekly lunch spending amounts fall between \$36.17 and \$53.03.

Would you like more details on the Empirical Rule or any clarifications?

### 5 Relative Questions:
1. What does it mean if the data is not normally distributed?
2. How is a z-score interpreted in normal distributions?
3. What is the significance of using the Empirical Rule?
4. Can the z-scores be used to find more precise percentages using z-tables?
5. How would the results change with a different standard deviation?

### Tip:
Understanding the Empirical Rule is crucial in quickly estimating the spread of data in normally distributed datasets. It helps you gauge how typical or extreme specific data points are.

In a recent survey of 655 working Americans ages 25-34, the average weekly amount spent on lunch was $44.6 with standard deviation $2.81. The weekly amounts are approximately bell-shaped. Estimate the percentage of amounts that are between $36.17 and $53.03.

Estimate the percentage of lunch spending amounts between $36.17 and $53.03 based on a normally distributed dataset of American workers aged 25-34. Learn how to use the Empirical Rule, z-scores, and key statistical formulas in this step-by-step solution.

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