N2021 P2 Q8

A car manufacturer claims that the front tyres on a particular model of car have an average life span of \(20\,000\) miles. Following comments from customers, the sales manager wishes to test if the life span of the tyres is greater than \(20\,000\) miles.

1. Explain why the sales manager should carry out a \(1\)-tail test. State hypotheses for the test, defining any symbols you use.[3]

The sales manager contacts customers and gathers details about the life spans of a random sample of \(50\) of these tyres. The life spans, \(x\) thousand miles, are summarised below.

\[\sum(x-20)=9.4\qquad\sum(x-20)^2=38.76\]

2. Calculate unbiased estimates of the population mean and variance of the life spans of the tyres.[2]
3. Test, at the \(5\%\) level of significance, whether the mean life span of front tyres is more than \(20\,000\) miles.[3]
4. Explain why this test would be inappropriate if the sales manager had taken a random sample of \(15\) tyres.[1]