Lunch break: In a recent survey of 655 working Americans ages 25 - 34, the average weekly amount spent on lunch was $42.68 with standard deviation $3.06. The weekly amounts are approximately bell-shaped.
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(a) Estimate the percentage of amounts that are between $33.50 and $51.86.
(Choose one)
of the amounts fall between $33.50 and $51.86.

To estimate the percentage of amounts that fall between $33.50 and $51.86, we can apply **the Empirical Rule** (also called the 68-95-99.7 Rule), since the distribution of weekly lunch expenses is approximately bell-shaped (normal distribution).

### Empirical Rule:
- About 68% of the data falls within 1 standard deviation of the mean.
- About 95% of the data falls within 2 standard deviations of the mean.
- About 99.7% of the data falls within 3 standard deviations of the mean.

#### Step-by-step process:

1. **Mean** = $42.68 (average weekly lunch amount)
2. **Standard deviation (σ)** = $3.06
3. Find how many standard deviations away the values $33.50 and $51.86 are from the mean.

#### Z-scores:
We calculate the z-scores for $33.50 and $51.86 using the formula:
\[
Z = \frac{X - \mu}{\sigma}
\]
Where $ X $ is the value, $ \mu $ is the mean, and $ \sigma $ is the standard deviation.

- **For $33.50**:
\[
Z = \frac{33.50 - 42.68}{3.06} \approx -3.00
\]

- **For $51.86**:
\[
Z = \frac{51.86 - 42.68}{3.06} \approx 3.00
\]

#### Interpretation:

The z-scores of -3.00 and +3.00 indicate that the values $33.50 and $51.86 are 3 standard deviations below and above the mean, respectively. According to the Empirical Rule, approximately **99.7%** of the values in a bell-shaped distribution fall within 3 standard deviations of the mean.

### Conclusion:
Approximately **99.7%** of the amounts fall between $33.50 and $51.86.

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Would you like more details on how the Empirical Rule works, or do you have any further questions?

Here are 5 related questions to explore further:
1. What does a z-score represent in a normal distribution?
2. How does the Empirical Rule help in real-world data analysis?
3. Can we apply the Empirical Rule to skewed data?
4. How would the percentage change for different intervals like within 1 or 2 standard deviations?
5. What is the importance of standard deviation in analyzing data sets?

**Tip:** Always check if the data is approximately normal (bell-shaped) before applying the Empirical Rule!

In a recent survey of 655 working Americans ages 25 - 34, the average weekly amount spent on lunch was $42.68 with a standard deviation of $3.06. The weekly amounts are approximately bell-shaped. Estimate the percentage of amounts that are between $33.50 and $51.86.

Learn how to estimate the percentage of values between $33.50 and $51.86 in a normally distributed data set with a mean of $42.68 and standard deviation of $3.06. This example applies the Empirical Rule and Z-scores to a real-world problem. Suitable for Grades 10-12.

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