Question

A company is developing a new high-performance wax for cross country ski racing. In order to justify the price marketing wants, the wax needs to be very fast. Specifically, the mean time to finish their standard test course should be less than 55 seconds for a former Olympic champion. To test it, the champion will ski the course 8 times. The champion's time (selected at random) 57.7, 63.9, 50.5, 50.5,48.8, 45.9, 54.2, and 42.3 seconds to complete the test course.

What is the test statistic?

What is the p value of the test statistic?

Should they market the wax?

Suppose they decide not to market the wax after the test, but it turns out that the wax really does lower the champion's average time to less than 55 seconds. What kind of error have they made? Explain the impact to the company of such an error.

Answer 1

Given the eight samples

First calculate the sample mean and the standard deviation.

Sample mean = = 51.725

Sample standard deviation s =   = 6.8

n = 8.

Let X be the time to finish their standard test course.

sample mean = 51.725

sample standard deviation s = 6.8

Let, = 0.05

n = 8 => degrees of freedom (df) = 7

We have to check if time to finish their standard test course is less than 55 or not.

As the population standard deviation is unknown but the sample standard deviation is known, so we will do a left-sided, one-sample t-test.

Standard error SE = s/ = 6.8/ = 2.4

Now,

t-statistic = ( ) / SE = (51.725-55) / 2.4 = -1.36

For t = -1.36 and df = 7, the one-sided p-value is 0.1

p > 0.05

We do not have strong evidence against the null hypothesis. So we fail to reject the null hypothesis and can claim with 95% confidence that the time to finish their standard test course is less than 55 seconds.

Hence they should market the wax.

Suppose they decide not to market the wax after the test, but it turns out that the wax really does lower the champion's average time to less than 55 seconds it means they failed to reject the null hypothesis when it is false.

This is a type II error. On marketing this wax, there will be an effect on the company in terms of profit they'll make by selling.