Practice section 5 of 5

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]

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

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

(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]

(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]

In this question you should state clearly the variables and the values of the parameters of any distribution you use.

A pharmaceutical company has 300 employees from the research department and 330 employees from the manufacturing department.

(i) Billy wishes to find out about the stress level of employees in the research department, so he sends a survey to the 300 employees in the research department. Explain whether these 300 employees form a sample or a population. [1]

(ii) Charlie wishes to conduct a survey regarding the employees’ experiences with a new facility.100 of them are randomly selected from the research department and the other 100 are randomly selected from the manufacturing department. Explain whether the method will form a random sample. [2]

The pharmaceutical company produces pills against a particular bacteria. The manager claims that the average time taken to manufacture a batch of pills is less than 12 hours. The time taken in hours, $x$, was measured on 80 occasions and the results are summarised as follows:

$\sum(x-12)=-12.8,\qquad\sum(x-12)^2=16.19.$

(iii) Find the unbiased estimates of the population mean and variance of the time taken to manufacture a batch of pills. [2]

(iv) Carry out a hypothesis test on the manager’s claim. Find the p-value for this test and explain what it indicates about the manager’s claim. [4]

It was found that the testing process had not been rigorous enough and the same hypothesis test was conducted in another random sample of 80 observations. The sample mean was found to be $k$ hours and the sample standard deviation was found to be 1.5 hours. It was found that the manager’s claim was not supported at 1% level of significance.

(v) Find the range of values of $k$, correct to 3 decimal places. [3]