Question

Suppose that the speed at which cars go on the freeway is normally distributed with mean 74 mph and standard deviation 5 miles per hour. Let X be the speed for a randomly selected car. Round all answers to 4 decimal places where possible.

a. What is the distribution of X? X ~ N(,)

b. If one car is randomly chosen, find the probability that it is traveling more than 73 mph.

c. If one of the cars is randomly chosen, find the probability that it is traveling between 77 and 80 mph.

d. 78% of all cars travel at least how fast on the freeway? mph.

Answer 1

Solution :

Given that ,

mean = = 74

standard deviation = = 5

a.

X N (74 , 5)

b.

P(x > 73) = 1 - P(x < 73)

= 1 - P[(x - ) / < (73 - 74) / )5

= 1 - P(z < -0.2)

= 1 - 0.4207

= 0.5793

Probability = 0.5793

c.

P(77 < x < 80) = P[(77 - 74)/ 5) < (x - ) /  < (80 - 74) / 5) ]

= P(0.6 < z < 1.2)

= P(z < 1.2) - P(z < 0.6)

= 0.8849 - 0.7257

= 0.1592

Probability = 0.1592

d.

Using standard normal table ,

P(Z > z) = 78%

1 - P(Z < z) = 0.78

P(Z < z) = 1 - 0.78

P(Z < -0.77) = 0.22

z = -0.77

Using z-score formula,

x = z * +

x = -0.77 * 5 + 74 = 70.15

fast on the freeway 70.15 mph