Separate a large-sample test from a known-variance test

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

$\sum(x-7)=24,\qquad\sum(x-7)^2=14908.$

(i) Explain in context, the meaning of a random sample of 120 patients.1 mark

(ii) Calculate the unbiased estimates of the population mean and variance of the recovery period of the patients.2 marks

(iii) Carry out a test at 5% significance level, on the claim that the average recovery period of 7 days is understated.4 marks

(iv) 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 mark

(v) 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, $\bar x$, such that the null hypothesis is rejected at 1% level of significance.4 marks