Question

Suppose that the speeds of cars travelling on California freeways are normally distributed with a mean of 61 miles/hour. The highway patrol's policy is to issue tickets for cars with speeds exceeding 75 miles/hour. The records show that exactly 1% of the speeds exceed this limit. Find the standard deviation of the speeds of cars travelling on California freeways. Carry your intermediate computations to at least four decimal places. Round your answer to at least one decimal place. miles/hour Clear Undo Help Next >> Explain

Answer 1

Answer:

The standard deviation of the speeds of cars travelling on California freeway is 6.0088 miles per hour.

Step-by-step explanation:

Problems of normally distributed samples can be 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

$Z = \frac{X - \mu}{\sigma}$

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 This p-value is the probability that the value of the measure is greater than X.

In this problem, we have that:

Suppose that the speeds of cars travelling on California freeways are normally distributed with a mean of 61 miles/hour. This means that $\mu = 61$.

The highway patrol's policy is to issue tickets for cars with speeds exceeding 75 miles/hour. The records show that exactly 1% of the speeds exceed this limit. This means that the pvalue of Z when $X = 75$ is 0.99. This is $Z = 2.33$

So

$Z = \frac{X - \mu}{\sigma}$

$2.33 = \frac{75 - 61}{\sigma}$

$2.33\sigma = 14$

$\sigma = \frac{14}{2.33}$

$\sigma = 6.0088$

The standard deviation of the speeds of cars travelling on California freeway is 6.0088 miles per hour.