HCI Sampling Tutorial Q8

The ‘reading age’ of children about to start a secondary school is a measure of how good they are at reading and understanding printed text. A child’s reading age, measured in years, is denoted by the random variable \(X\). The distribution \(X\) is assumed to be \(\mathrm{N}\left( \mu ,{{\sigma }^{2}} \right)\).

1. The reading age of a random sample of \(20\) children were measured, and the data obtained is summarised by \(\sum{x=232.6}\), \(\sum{{{x}^{2}}=2756.22}\).
   Calculate unbiased estimates of \(\mu \) and \({{\sigma }^{2}}\), giving your answers correct to \(2\) decimal places.
2. In order to obtain a more accurate estimate of \(\mu \), it is proposed to take a larger sample. Estimate the sample size needed so that we can be \(95\%\) certain that the sample mean reading age will differ from the true mean reading age by less than \(6\) months. Assume that it is known from previous studies that \({{\sigma }^{2}}=2.25\).