A radar unit is used to measure the speed of automobiles on an expressway during

Question

A radar unit is used to measure the speed of automobiles on an expressway during rush-hour traffic. The speeds of individual automobiles are normally distributed with a mean of 64.5 mph. (Give your answers correct to two decimal places.)

(a) Find the standard deviation of all speeds if 5% of the automobiles travel faster than 71 mph.

(b) Using the standard deviation found in part (a), find the percentage of these cars that are traveling less than 55 mph. %

(c) Using the standard deviation found in part (a), find the 99th percentile for the variable "speed."

Explanation / Answer

solution=

a. Find the mean of all speeds if 5% of the automobiles travel faster than 71 mph. Round the mean to the nearest tenth.
Mark a point on the horizontal axis with a right-tail of 5%.
Mark the point as x = 71.
Find the z-value that has a right tail of 5%:
invNorm(0.95) = 1.96
Now find "std":
x = z*s + u
71 = 1.96* s+ 64.5
std = 3.32 mph when rounded

(b) Using the standard deviation found in part (a), find the percentage of these cars that are traveling less than 55 mph

p( x< 55) = 55 - 64.5 / 3.32= -2.86

p( z <2.625) = 0.002118= 0.21%

(c) Using the standard deviation found in part (a), find the 99th percentile for the variable "speed

Find the z-value with of 99%
invNorm(0.99) = 2.58
Find the corresponding x value:
x = z*s+u = 2.58 *3.32 +64.5 = 73.07 mph

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