OCR MEI S2 (Statistics 2) 2013 June

1 Salbutamol is a drug used to improve lung function. In a medical trial, a random sample of 60 people with impaired lung function was selected. The forced expiratory volume in one second (FEV1) was measured for each person, both before being given salbutamol and again after a two-week course of the drug. The variables \(x\) and \(y\), measured in suitable units, represent FEV1 before and after the two-week course respectively. The data are illustrated in the scatter diagram below, together with the summary statistics for these data.
\includegraphics[max width=\textwidth, alt={}, center]{f3690bc0-3392-4f29-86f7-797d33fab4f1-2_682_1024_502_516} Summary statistics: $$n = 60 , \quad \sum x = 43.62 , \quad \sum y = 55.15 , \quad \sum x ^ { 2 } = 32.68 , \quad \sum y ^ { 2 } = 51.44 , \quad \sum x y = 40.66$$

1. Calculate the sample product moment correlation coefficient.
2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is positive correlation between FEV1 before and after the course.
3. State the distributional assumption which is necessary for this test to be valid. State, with a reason, whether the assumption appears to be valid.
4. Explain the meaning of the term 'significance level'.
5. Calculate the values of the summary statistics if the data point \(x = 0.55 , y = 1.00\) had been incorrectly recorded as \(x = 1.00 , y = 0.55\).

2 Suppose that 3\% of the population of a large city have red hair.

1. A random sample of 10 people from the city is selected. Find the probability that there is at least one person with red hair in this sample. A random sample of 60 people from the city is selected. The random variable \(X\) represents the number of people in this sample who have red hair.
2. Explain why the distribution of \(X\) may be approximated by a Poisson distribution. Write down the mean of this Poisson distribution.
3. Hence find
   (A) \(\mathrm { P } ( X = 2 )\),
   (B) \(\mathrm { P } ( X > 2 )\).
4. Discuss whether or not it would be appropriate to model \(X\) using a Normal approximating distribution. A random sample of 5000 people from the city is selected.
5. State the exact distribution of the number of people with red hair in the sample.
6. Use a suitable Normal approximating distribution to find the probability that there are at least 160 people with red hair in the sample.

3 The scores, \(X\), in Paper 1 of an English examination have an underlying Normal distribution with mean 76 and standard deviation 12. The scores are reported as integer marks. So, for example, a score for which \(75.5 \leqslant X < 76.5\) is reported as 76 marks.

1. Find the probability that a candidate's reported mark is 76 .
2. Find the probability that a candidate's reported mark is at least 80 .
3. Three candidates are chosen at random. Find the probability that exactly one of these three candidates' reported marks is at least 80 . The proportion of candidates who receive an A* grade (the highest grade) must not exceed \(10 \%\) but should be as close as possible to \(10 \%\).
4. Find the lowest reported mark that should be awarded an A* grade. The scores in Paper 2 of the examination have an underlying Normal distribution with mean \(\mu\) and standard deviation 12.
5. Given that \(20 \%\) of candidates receive a reported mark of 50 or less, find the value of \(\mu\).

4 An art gallery is holding an exhibition. A random sample of 150 visitors to the exhibition is selected. The visitors are asked which of four artists they prefer. Their preferences, classified according to whether the visitor is female or male, are given in the table.

1. Carry out a test at the \(10 \%\) significance level to examine whether there is any association between artist preferred and sex of visitor. Your working should include a table showing the contributions of each cell to the test statistic.
2. For each artist, comment briefly on how the preferences of each sex compare with what would be expected if there were no association.