Question

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 48 mph and the standard deviation was 17 mph. A histogram revealed that the vehicle speed at impact distribution was approximately normal. (Use the Empirical Rule.)

Answer 1

Explanation:

According to the Empirical rule:

• 68% of the distribution lie within σ of μ.

• 95% of the distribution lie within 2σ of μ.

• 99.7% of the distribution lie within 3σ of μ.

Given:

μ = 48 mph

σ = 17 mph

(1)

Compute the value within which, 68% of the distribution lie as follows:

`P(x_(1)<X<x_(2))=0.68\\P(\mu-\sigma<X<\mu+\sigma)=0.68\\`

The limits are:

`\mu-\sigma=48-17=31\\\mu+\sigma=48+17=65`

Thus, according to the Empirical 68% of the distribution of car speed lie between 31 mph and 65 mph.

(2)

Compute the value within which, 95% of the distribution lie as follows:

`P(x_(1)<X<x_(2))=0.95\\P(\mu-2\sigma<X<\mu+2\sigma)=0.95\\`

The limits are:

`\mu-2\sigma=48-(2*17)=14\\\mu+2\sigma=48+(2*17)=82`

Thus, according to the Empirical 95% of the distribution of car speed lie between 14 mph and 82 mph.

(3)

Compute the value within which, 99.7% of the distribution lie as follows:

`P(x_(1)<X<x_(2))=0.997\\P(\mu-3\sigma<X<\mu+3\sigma)=0.997\\`

The limits are:

`\mu-3\sigma=48-(*17)=-3\approx0\\\mu+3\sigma=48+(3*17)=99`

Thus, according to the Empirical 99.7% of the distribution of car speed lie between 0 mph and 99 mph.