In a study investigating the effect of car speed on accident severity, 5,000 reports of fatal automobile accidents were examined, and the vehicle speed at impact was recorded for each one. For these 5,000 accidents, the average speed was 47 mph and the standard deviation was 16 mph. A histogram revealed that the vehicle speed at impact distribution was approximately normal. (Use the Empirical Rule.)
(a)
Approximately what percentage of these vehicle speeds were between 31 and 63 mph?
approximately 
 %
(b)
Approximately what percentage of these vehicle speeds exceeded 63 mph? (Round your answer to the nearest whole number.)
approximately 
 %

Given the data in the problem, we will use the **Empirical Rule** (also known as the 68-95-99.7 rule) which applies to normal distributions. This rule states that:

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

The mean (\(\mu\)) is given as 47 mph and the standard deviation (\(\sigma\)) is 16 mph.

### Part (a): Percentage of vehicle speeds between 31 and 63 mph

We can calculate how many standard deviations 31 mph and 63 mph are from the mean.

1. **For 31 mph:**
   \[
   z = \frac{31 - 47}{16} = \frac{-16}{16} = -1
   \]
   This means 31 mph is 1 standard deviation below the mean.

2. **For 63 mph:**
   \[
   z = \frac{63 - 47}{16} = \frac{16}{16} = 1
   \]
   This means 63 mph is 1 standard deviation above the mean.

According to the Empirical Rule, **68% of the data** falls within 1 standard deviation of the mean, so approximately **68%** of the vehicle speeds are between 31 mph and 63 mph.

### Part (b): Percentage of vehicle speeds exceeding 63 mph

We already calculated that 63 mph is 1 standard deviation above the mean. According to the Empirical Rule:

- 68% of the data falls between 31 mph and 63 mph.
- The remaining 32% of the data is outside this range, with half of it (16%) above 63 mph.

Thus, approximately **16%** of the vehicle speeds exceed 63 mph.

### Final Answers:
- (a) Approximately **68%** of the vehicle speeds were between 31 mph and 63 mph.
- (b) Approximately **16%** of the vehicle speeds exceeded 63 mph.

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

Here are 5 related questions to explore:

1. How does the Empirical Rule apply to skewed distributions?
2. What if the standard deviation were smaller? How would the percentages change?
3. Can the normal distribution be used for non-fatal accidents in the same way?
4. How do z-scores help in finding probabilities for normal distributions?
5. How do you calculate probabilities for values not covered by the Empirical Rule?

**Tip**: Always double-check if the distribution is approximately normal before applying the Empirical Rule!

In a study investigating the effect of car speed on accident severity, 5,000 reports of fatal automobile accidents were examined, and the vehicle speed at impact was recorded for each one. For these 5,000 accidents, the average speed was 47 mph and the standard deviation was 16 mph. A histogram revealed that the vehicle speed at impact distribution was approximately normal. (Use the Empirical Rule.) (a) Approximately what percentage of these vehicle speeds were between 31 and 63 mph? (b) Approximately what percentage of these vehicle speeds exceeded 63 mph?

Analyze a real-world problem using the Empirical Rule. Given a dataset of 5,000 fatal accidents with an average speed of 47 mph and a standard deviation of 16 mph, learn how to apply the Empirical Rule to calculate the percentage of vehicle speeds between 31 mph and 63 mph, and those exceeding 63 mph. This problem is ideal for high school students studying statistics.

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