Question

The distribution of passenger vehicle speeds traveling on a certain freeway in California is nearly normal with a mean of 73.7 miles/hour and a standard deviation of 4.72 miles/hour. The speed limit on this stretch of the freeway is 70 miles/hour.

(a) A highway patrol officer is hidden on the side of the freeway. What is the probability that 5 cars pass and none are speeding?
Assume that the speeds of the cars are independent of each other. (Round your answer to four decimal places.)
(b) On average, how many cars would the highway patrol officer expect to watch until the first car that is speeding?
(c) What is the standard deviation of the number of cars he would expect to watch?

Answer 1

a) 0.2347, 0.0007

b) 1.30

c) 0.63

Explanation:

Given –

Mean = 73.4

Sigma = 4.7

Probability that the speed is greater than 70 is

P (X>70) = P (z> (70-73.4)/4.7) = P (z>-0.72) = 0.7653

a) The probability that a car is not speeding = 1 - 0.7653 = 0.2347

The probability that all 5 cars are not speeding = (0.2347)5 = 0.0007

b) E (X) = 1/p = 1/0.7653 = 1.30

c) Standard deviation = Sqrt (1-p/p) = (1-0.7653)/0.7653 = 0.63