Question

Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph. 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

Based on this criterion, 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 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?

X when Z has a pvalue of 1-0.1 = 0.9.

So X when Z = 1.28.

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

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

`X - 71 = 8*1.28`

`X = 81.24`

Based on this criterion, the new speed limit will be 81.24 mph.