Answer:
a) 22.66% of vehicles are less than or equal to the speed limit
b) So 0.47% of the vehicles would be going less than 50 mph
c) The new speed limit will be 81.24 mph.
Explanation:
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:

a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?
This is the pvalue of Z when X = 65. So



has a pvalue of 0.2266.
So 22.66% of vehicles are less than or equal to the speed limit
b. What proportion of the vehicles would be going less than 50 mph?
This is the pvalue of Z when X = 50. So



has a pvalue of 0.0047.
So 0.47% of the vehicles would be going less than 50 mph
c. A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion?
This is the value of X when Z has a pvalue of 1-0.1 = 0.9. So it is X when




