A radar unit is used to measure the speed of automobile on an expressway during
ID: 3158888 • Letter: A
Question
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".
Explanation / Answer
a)
For a right tailed area of 0.03, the z score is, by table/technology,
z = 1.880793608
Hence, as x = u + z*sigma,
72 = u + 1.880793608*4
Thus,
u = 64.47682557 mph [ANSWER]
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b)
We first get the z score for the two values. As z = (x - u) / s, then as
x1 = lower bound = 70
x2 = upper bound = 75
u = mean = 64.47682557
s = standard deviation = 4
Thus, the two z scores are
z1 = lower z score = (x1 - u)/s = 1.380793608
z2 = upper z score = (x2 - u) / s = 2.630793608
Using table/technology, the left tailed areas between these z scores is
P(z < z1) = 0.916328785
P(z < z2) = 0.995740712
Thus, the area between them, by subtracting these areas, is
P(z1 < z < z2) = 0.079411927 [ANSWER]
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c)
First, we get the z score from the given left tailed area. As
Left tailed area = 0.25
Then, using table or technology,
z = -0.67448975
As x = u + z * s,
where
u = mean = 64.47682557
z = the critical z score = -0.67448975
s = standard deviation = 4
Then
x = critical value = 61.77886657 mph [ANSWER]
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