Question

Suppose that a population of brakes supplied has a mean stopping distance, when the brake is applied fully to a vehicle traveling at 75 mph is 268 feet. Population standard deviation is 20 feet. Suppose that you take a sample of n brakes to test and if the average stopping distance is less than or equal to a critical value, you accept the lot. If it is more than the critical value, you reject the lot. You want an alpha of 0.03. Further, we want to reject lots with a population mean stopping distance of 282 ft., we want to reject with a probability of 0.9. (Note that it is one sided, since we would not get worried if the vehicle stops at a distance shorter than the average time.)

1. Draw a diagram and mark the two distributions, alpha, beta and CV.

2. What is the correct sample size that can achieve this?

3. What is the critical value?

Answer 1

It is given that

The null hypothesis is

The alternate hypothesis is

the significance level for the test is

we want to reject the null hypothesis with probability 0.9 when the average stopping distance is 282.

that means the power of the test is 0.9 given

a)

is power, the probability of detecting a significant result, which is equal =0.9

CV is the coefficient of variation

b) formula for calculating the sample size when power of the test and significance level is known

is the effect size

the correct sample size that can achieve this is 49

c)  This is a right tailed test

We will fail to reject the null (commit a Type II error) if we get a Z statistic less than 1.8808

This 1.8808 Z-critical value corresponds to some X critical value ( X critical), such that

Solving