2024 ACJC P2 Q8

A refrigerator manufacturer claims that the mean lifespan of refrigerators of a particular model is \(12\) years. A consumers association representative suspects that the mean lifespan of the refrigerators is actually less than \(12\) years.

The durations \(x\), in years, of a random sample of \(45\) refrigerators are summarised as follows.

\[\sum(x-12) = -4.3 \qquad \sum(x-12)^2 = 17.08\]

1. Calculate unbiased estimates of the population mean and variance of the lifespan of the refrigerators.[2]
2. State hypotheses that can be used to test if the mean lifespan of the refrigerators is less than \(12\) years, defining any parameters you use. Test, at the \(5\%\) level of significance, whether the mean lifespan of the refrigerators is less than \(12\) years.[4]

The manufacturer switches to a different coolant and decides to test whether the mean lifespan of the refrigerators has changed from \(12\) years. He records the durations of a large random sample of \(n\) refrigerators and finds that their mean lifespan is \(12.4\) years and variance is \(4.1\) square years.

A test at \(5\%\) significance level, is carried out on the new random sample. The test shows that there is sufficient evidence that the mean lifespan of refrigerators has changed.

4. Find the range of possible values of \(n\).[4]