Unbiased estimator from summary statistics

3 The lengths, \(x \mathrm {~mm}\), of a random sample of 150 insects of a certain kind were found. The results are summarised by \(\Sigma x = 7520\) and \(\Sigma x ^ { 2 } = 413540\).

1. Calculate unbiased estimates of the population mean and variance of the lengths of insects of this kind.
2. Using the values found in part (i), calculate an estimate of the probability that the mean length of a further random sample of 80 insects of this kind is greater than 53 mm .

4 A set of observations of a random variable \(W\) can be summarised as follows: $$n = 14 , \quad \Sigma w = 100.8 , \quad \Sigma w ^ { 2 } = 938.70 .$$

1. Calculate an unbiased estimate of the variance of \(W\).
2. The mean of 70 observations of \(W\) is denoted by \(\bar { W }\). State the approximate distribution of \(\bar { W }\), including unbiased estimate(s) of any parameter(s).

1 A random sample of observations of a random variable \(X\) is summarised by $$n = 100 , \quad \Sigma x = 4830.0 , \quad \Sigma x ^ { 2 } = 249 \text { 509.16. }$$

1. Obtain unbiased estimates of the mean and variance of \(X\).
2. The sample mean of 100 observations of \(X\) is denoted by \(\bar { X }\). Explain whether you would need any further information about the distribution of \(X\) in order to estimate \(\mathrm { P } ( \bar { X } > 60 )\). [You should not attempt to carry out the calculation.]

7 A random sample of 50 full-time university employees was selected as part of a higher education salary survey. The annual salary in thousands of pounds, \(x\), of each employee was recorded, with the following summarised results. $$\sum x = 2290.0 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 28225.50$$ Also recorded was the fact that 6 of the 50 salaries exceeded \(\pounds 60000\).

1. 1. Calculate values for \(\bar { x }\) and \(s\), where \(s ^ { 2 }\) denotes the unbiased estimate of \(\sigma ^ { 2 }\).
   2. Hence show why the annual salary, \(X\), of a full-time university employee is unlikely to be normally distributed. Give numerical support for your answer.
2. 1. Indicate why the mean annual salary, \(\bar { X }\), of a random sample of 50 full-time university employees may be assumed to be normally distributed.
   2. Hence construct a \(99 \%\) confidence interval for the mean annual salary of full-time university employees.
3. It is claimed that the annual salaries of full-time university employees have an average which exceeds \(\pounds 55000\) and that more than \(25 \%\) of such salaries exceed £60000. Comment on each of these two claims.

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).

The times, \(T\) minutes, taken by a random sample of \(75\) students to complete a test were noted. The results were summarised by \(\sum t = 230\) and \(\sum t^2 = 930\).

1. Calculate unbiased estimates of the population mean and variance of \(T\). [3]

You should now assume that your estimates from part (a) are the true values of the population mean and variance of \(T\).

1. The times taken by another random sample of \(75\) students were noted, and the sample mean, \(\overline{T}\), was found. Find the value of \(a\) such that \(P(\overline{T} > a) = 0.234\). [3]