In a study investigating the effect of car speed on accident severity, the vehicle speed at impact was
recorded for 5,000 fatal accidents. For these accidents, the mean speed was 47 mph and the standard
deviation was 17 mph. A histogram revealed that the vehicle speed distribution was mound shaped and
approximately symmetric. (Use the Empirical Rule.)
(a) Approximately what percentage of vehicle speeds were between 30 and 64 mph?
(b) Approximately what percentage of vehicle speeds exceeded 64 mph? (Round your answer to the
nearest whole number.)
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(a) To find the percentage of vehicle speeds between 30 and 64 mph, we need to find the Z-scores for these speeds and use the empirical rule.
First, we need to find the Z-score for the lower speed of 30 mph:
Z = (X - μ) / σ
Z = (30 - 47) / 17
Z ≈ -1
Second, we need to find the Z-score for the higher speed of 64 mph:
Z = (X - μ) / σ
Z = (64 - 47) / 17
Z ≈ 1
According to the empirical rule:
- Approximately 68% of the data falls within 1 standard deviation of the mean.
- Approximately 95% of the data falls within 2 standard deviations of the mean.
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
Since the speeds are approximately symmetric and mound shaped, we can estimate that the percentage of vehicle speeds between 30 and 64 mph is approximately:
68% + 95% = 163%
(b) To find the percentage of vehicle speeds that exceeded 64 mph, we look at the Z-score for 64 mph:
Z = (X - μ) / σ
Z = (64 - 47) / 17
Z ≈ 1
From the empirical rule, we know that approximately 68% of the data falls within 1 standard deviation of the mean. This means that approximately 16% falls below 1 standard deviation below the mean, and approximately 16% falls above 1 standard deviation above the mean.
Therefore, the percentage of vehicle speeds that exceeded 64 mph is approximately:
16% + 16% = 32%
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