Question

Assume the speed of vehicles along the highway has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.

a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?
b. What proportion 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?

Answer 1

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
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

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:

`\mu= 71, \sigma = 8`

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

`Z = (X - \mu)/(\sigma)`

`Z = (65 - 71)/(8)`

`Z = -0.75`

`Z = -0.75` 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

`Z = (X - \mu)/(\sigma)`

`Z = (50 - 71)/(8)`

`Z = -2.6`

`Z = -2.6` 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
`Z = 1.28`

`Z = (X - \mu)/(\sigma)`

`1.28 = (X - 71)/(8)`

`X - 71 = 8*1.28`

`X = 81.24`