In: Statistics and Probability
The speed limit is 25 mph on Cumberland Avenue. During the construction along Cumberland Avenue, assume that the speed of cars follows the normal model with mean 22.53 mph and a standard deviation of 2.47 mph.
a) Find the z-score for the speed limit.
b) What percent of cars would you expect to be going over the speed limit?
c) If a car is captured on radar at 28 mph, would you consider the speed of the car unusual? Justify your answer.
d) What percent of cars would be driving with 28 mph or slower? Using only the 68-95-99.7 rule, provide as narrow of an interval as you can that contains the answer to this question. Fill in the blanks below.
The answer is greater than ________% but less than ________ %
a)
Answer:
z score = 1
Explanation:
The z score is obtained using the following formula,
b)
Answer:
We can expect 15.87% of cars going over the speed limit.
Explanation:
The probability for the corresponding z score is obtained from the z distribution table (In excel use function =1-NORM.S.DIST(1,TRUE))
c)
Answer:
Yes, the speed of 28 mph is unusual
Explanation:
Since the speed of 28 mph is farther from the mean compared to the 25 mph, the z score will be larger and thus the percent of cars above the mean speed will be smaller hence the speed of the car will be considered unusual. The probability calculation is shown below,
There are only 1.34% of cars are above 28 mph which is a small number.
d)
Answer:
The answer is greater than 95% and less than 99.7%.
Explanation:
The 68-95-99.7 rule says that approximately,
68% of the data value lies within one standard deviation from the mean,
95% of the data value lies within two standard deviations from the mean,
99.7% of the data value lies within three standard deviations from the mean,
The z score for the speed of 28 mph is,
The speed of 28 mph is greater than 2 standard deviations but less than 3 standard deviation from the mean speed = 22.53 mph. Now using the 68-95-99.7 rule we can say that approximately greater than 95% and less than 99.7% percent of cars would be driving with 28 mph or slower.
The z distribution table
For part b)
For part c)