2008 MJC MYE P2 Q11

A gaming company claims that gamers spend an average of \(m\) hours daily on a new game that they have launched. A random sample of \(30\) gamers was surveyed and the time, \(x\) hours, spent daily on the game was recorded. The results are summarized by

\(\sum{x}=126\) , \(\sum{{{x}^{2}}}=550.1525\).

1. Calculate the unbiased estimates of the population mean and variance of the number of hours a gamer spends daily on the game.[2]

A test is carried out, at the \(5\%\) significance level, to determine whether the average time spent daily on the game has been overstated by the company.

2. State appropriate hypotheses for the test.[1]
3. Find the range of values of \(m\) for which the null hypothesis is not rejected.[3]

A second sample of \(50\) gamers is surveyed and the time, \(y\) hours, spent daily on the game was recorded, and the results are summarized by

\(\sum{y}=215\),\(\sum{{{y}^{2}}}=964.19\).

4. The two samples are combined into a single sample and a test is carried out at the \(5\%\) level of significance level to determine whether the company’s claim is valid. Given that the value of \(m\) is \(4.5\), find \(p\)-value of this test.[4]