Question #13309

A radar unit is used to measure the speed of automobile on an expressway during rush-hour traffic. The speeds of individual automobiles are normally distributed with a standard deviation of 4 mph.
a. Find the mean of all speeds if 3% of the automobiles travel faster than 72 mph. Round the mean to the nearest tenth.
b. Using the mean you found in part a, find the probability that a car is traveling between 70 mph and 75 mph. Interpret the meaning of this answer.
c. Using the mean you found in part a, find the 25th percentile for the variable "speed".

Expert's answer

a. Find the mean of all speeds if 3% of the automobiles travel faster than 72 mph. Round the mean to the nearest tenth.

Mark a point on the horizontal axis with a right-tail of 3%.

Mark the point as x = 72.

Find the z-value that has a right tail of 3%:

invNorm(0.97) = 1.8808

Now find "u":

x = z*s + u

72 = 1.8808*4 + u

u = 64.4768 = 64.5 mph when rounded

b. Using the mean you found in part a, find the probability that a car is traveling between 70 mph and 75 mph. Interpret the meaning of this answer.

z(70) = (70-64.5)/4 = 1.375

z(75) = (75-64.5)/4 = 2.625

P(70<= x <=75) = P(1.375<= z <=2.625) = 0.08

c. Using the mean you found in part a, find the 25th percentile for the variable "speed".

Find the z-value with a left-tail of 25%

invNorm(0.25) = -0.6745

Find the corresponding x value:

x = zs+u = -0.6745*4+64.5 = 61.8 mph

Mark a point on the horizontal axis with a right-tail of 3%.

Mark the point as x = 72.

Find the z-value that has a right tail of 3%:

invNorm(0.97) = 1.8808

Now find "u":

x = z*s + u

72 = 1.8808*4 + u

u = 64.4768 = 64.5 mph when rounded

b. Using the mean you found in part a, find the probability that a car is traveling between 70 mph and 75 mph. Interpret the meaning of this answer.

z(70) = (70-64.5)/4 = 1.375

z(75) = (75-64.5)/4 = 2.625

P(70<= x <=75) = P(1.375<= z <=2.625) = 0.08

c. Using the mean you found in part a, find the 25th percentile for the variable "speed".

Find the z-value with a left-tail of 25%

invNorm(0.25) = -0.6745

Find the corresponding x value:

x = zs+u = -0.6745*4+64.5 = 61.8 mph

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