2025 VJC P2 Q12

A hospital introduces a new medication regimen to help patients recover from a viral infection more quickly. Previously, the average recovery time was \(7.2\) days.

After adopting the new regimen, a random sample of \(40\) patients is observed, and their recovery times, \(x\), in days, are recorded. The results are summarised below. \[\sum(x - 7.2) = -42 \quad \text{and} \quad \sum(x - 7.2)^2 = 648\]

1. Explain whether the hospital should carry out a one-tailed test or a two-tailed test.[1]
2. Calculate unbiased estimates of the population mean and variance of the recovery times under the new regimen.[2]
3. Carry out an appropriate test at the \(5\%\) significance level. You should state clearly the hypotheses for your test and define any parameters that you use.[4]
4. Explain, in the context of the question, the meaning of 'at the \(5\%\) significance level'.[1]
5. Explain whether it would have been sufficient for the hospital to take a random sample of \(10\) patients and record their recovery times under the new regimen in order to carry out the hypothesis test.[1]

It was discovered that an intern at the hospital had made an error in recording the recovery times. A random sample of another \(30\) patients was taken and their recovery times under the new regimen was taken and recorded by a senior researcher. The sample standard deviation was found to be \(1.7\) days and the sample mean time was denoted by \(k\).

6. A test at the \(1\%\) significance level concludes that the average recovery time did not improve from \(7.2\) days. Find the set of values of \(k\).[3]