Review practice · Section 2 of 2

A manufacturing process produces ball bearings with diameters with known standard deviation 0.04 cm. Under normal circumstances, the manufacturing process will produce ball bearings of mean diameter 0.5 cm.

(i) During a routine quality control check, a random sample of 25 ball bearings gives a mean of 0.51 cm. Is there evidence to believe that the manufacturing process is producing ball bearings of the stated diameter? Perform an appropriate test at 5% level of significance. State a necessary assumption for the test to be valid.4 marks

An enhancement on the manufacturing process will ensure that the diameters of the ball bearings produced are less variable.

(ii) Measurements of a sample of 100 ball bearings give the following summary statistics:

$\sum x=50.6,\qquad\sum(x-0.5)^2=0.08345.$

Show that the unbiased estimate of the population variance is $8.07\times 10^{-4}$.

Is there evidence at the 5% level of significance that after the enhancement, the manufacturing process is producing oversized ball bearings?4 marks

(iii) Another sample of 100 ball bearings yield the same summary statistics as the previous sample in (ii). Explain, with justification, whether the combined sample will give a different conclusion to (ii).4 marks

(a) College students intending to further their studies overseas have to sit for a mandatory Overseas Universities Test (OUT). Researcher Mr Anand wishes to find out if male college students tend to score higher for OUT compared to female college students.

Mr Anand’s colleague randomly selects 150 male and 150 female students from the combined student population of three particular colleges near his home to form a sample of 300 college students for the research.

Explain whether this sample is a random sample.2 marks

(b) The mean OUT score for all college students in 2016 is 66. Mr Anand randomly selects 240 college students taking OUT in 2017 and their scores, $x$, are summarised in the following table:

Score, $x$	60	65	68	70	75	80
Frequency, $f$	40	90	63	27	18	2

(i) Write down the unbiased estimates of the population mean and variance of the OUT scores for the college students in 2017.1 mark

(ii) Test, at the 10% level of significance, whether the mean OUT score for all college students in 2017 is higher than the mean score attained in 2016.4 marks

(iii) Explain what is meant by the phrase “10% level of significance” in this context.1 mark

(iv) Mr Anand draws a new sample of 240 male college students. Using the unbiased estimate for the population variance computed in (i), find the range of values for the sample mean $\bar x$ that is required for this new sample to achieve a different conclusion from that in (ii).4 marks

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

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 mark

(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 marks

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 marks

(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 marks

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 marks