Answer to Question #189150 in Quantitative Methods for Ahmad

Question

A prototype automotive tire has a design life of 38,500 miles with a standard deviation of 2,500 miles. Five such tires are manufactured and tested. On the assumption that the actual population mean is 38,500 miles and the actual population standard deviation is 2,500 miles, find the probability that the sample mean will be less than 36,000 miles. Assume that the distribution of lifetimes of such tires is normal.

Answer 1

For simplicity we use units of thousands of miles. Then the sample mean $\bar X$ has mean

$\mu_{\bar x}=\mu=38.5$ and standard deviation $\frac{\sigma_{\bar x}}{\sqrt n}=\frac{2.5}{\sqrt 5}=1.11803$ . Since the population is normally distributed, so is $\bar X$ hence

$$P(\bar X<36)=P(Z<\frac{36-\mu_{\bar X}}{\sigma_{\bar X}})\\\Rightarrow P(Z< \frac{36-38.5}{1.11803})\\\Rightarrow P(Z<-2.24)=0.0125$$

That is, if the tires perform as designed, there is only about a 1.25% chance that the average of a sample of this size would be so low.