7. A telephone company believes that, for young people, the average length of a telephone call on a land line is longer than on a mobile, due to the difference in price. The company collected data on the time, \(t\) minutes, of 500 calls made by young people on mobiles and the data is summarised by $$\Sigma t = 7335 , \quad \Sigma t ^ { 2 } = 172040 .$$

1. Calculate unbiased estimates of the mean and variance of \(t\). For 200 calls made on land lines by the same young people, unbiased estimates of the mean and variance of the call length were 15.9 minutes and 108.5 minutes \({ } ^ { 2 }\) respectively.
2. Stating your hypotheses clearly, test at the \(5 \%\) level whether or not there is evidence that longer calls are made on land lines than on mobiles.
   (9 marks)
3. Explain the importance of the central limit theorem in carrying out the test in part (b).