Question 11 (16 points). The distribution of automobile speeds on a certain high

Question

Question 11 (16 points). The distribution of automobile speeds on a certain highway is normally distributed with a mean of 71.6 miles/hour, and a standard deviation of 4.53 miles/hour. (a) What is the probability that an automobile travels slower than 75 miles/hour? (b) What is the probability that an automobile travels faster than 65 miles/hour? (e) What is the probability that an (d) What is the 90th percentile for the speeds of automobiles on this highway? automobile travels between 65 and 75 miles/hour?

Explanation / Answer

First we write down the given data as shown below:

Mean, m = 71.6

Standard deviation, S = 4.53

(a)

At X = 75, we first calculate the z-score:

z = (X-m)/S = (75-71.6)/4.53 = 0.75

Using a cumulative z-table, we see that:

P(X < 75) = P(z < 0.75) = 0.77

(b)

At X = 65, we first calculate the z-score:

z = (X-m)/S = (65-71.6)/4.53 = -1.46

Using a cumulative z-table, we see that:

P(X > 65) = P(z > -1.46) = 0.93

(c)

Using the values calculated in the above two parts:

P(65 < X < 75) = P(X < 75) - P(X < 65) = P(X < 75) - (1-P(X > 65)) = 0.77-(1-0.93) = 0.70

(d)

For the 90th percentile, the right tailed p-value is:

p = 0.10

The corresponding z-score for this p-value is:

z = 1.282

So,

X90 = z*S + m = 1.282*4.53 + 71.6 = 77.407 mph

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