Audit the sampling frame and a frequency-table test

(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