2022 CJC P2 Q12

A virologist claims that due to the introduction of vaccine, the recovery period from a certain type of flu-like virus is on average \(7\) days. However, an analyst who works with such patients on a day-to-day basis believes that the average recovery period quoted by the virologist is understated. He carries out a survey to investigate this by recording the recovery period of \(120\) randomly chosen patients. The recovery period of each of these \(120\) patients are denoted by \(x\) days. The summarised results are given by

\(\sum{\left( x-7 \right)=24}\) and \(\sum{{{\left( x-7 \right)}^{2}}=\mathrm{14908}}\)

1. Explain in context, the meaning of a random sample of \(120\) patients.[1]
2. Calculate the unbiased estimates of the population mean and variance of the recovery period of the patients.[2]
3. Carry out a test at \(5\%\) significance level, on the claim that the average recovery period of \(7\) days is understated.[4]
4. Explain if the test in part (iii) would still be valid if the analyst had asked for recovery periods from a random sample of \(15\) patients.[1]
5. Suppose instead, the analyst wants to test if the average recovery period differs from \(7\) days. The analyst collects data from another random sample of \(120\) patients. Given that the population standard deviation is now known to be \(1\) day, find the range of values of the sample mean, \(\overline{x}\), such that the null hypothesis is rejected at \(1\%\) level of significance.[4]