OCR MEI S2 (Statistics 2) 2007 June

1 The random variable \(X\) represents the time taken in minutes for a haircut at a barber's shop. \(X\) is Normally distributed with mean 11 and standard deviation 3 .

1. Find \(\mathrm { P } ( X < 10 )\).
2. Find the probability that exactly 3 out of 8 randomly selected haircuts take less than 10 minutes.
3. Use a suitable approximating distribution to find the probability that at least 50 out of 100 randomly selected haircuts take less than 10 minutes. A new hairdresser joins the shop. The shop manager suspects that she takes longer on average than the other staff to do a haircut. In order to test this, the manager records the time taken for 25 randomly selected cuts by the new hairdresser. The mean time for these cuts is 12.34 minutes. You should assume that the time taken by the new hairdresser is Normally distributed with standard deviation 3 minutes.
4. Write down suitable null and alternative hypotheses for the test.
5. Carry out the test at the \(5 \%\) level.

2 A medical student is trying to estimate the birth weight of babies using pre-natal scan images. The actual weights, \(x \mathrm {~kg}\), and the estimated weights, \(y \mathrm {~kg}\), of ten randomly selected babies are given in the table below.

1. Calculate the value of Spearman's rank correlation coefficient.
2. Carry out a hypothesis test at the \(5 \%\) level to determine whether there is positive association between the student's estimates and the actual birth weights of babies in the underlying population.
3. Calculate the value of the product moment correlation coefficient of the sample. You may use the following summary statistics in your calculations: $$\Sigma x = 31.63 , \quad \Sigma y = 33.1 , \quad \Sigma x ^ { 2 } = 101.92 , \quad \Sigma y ^ { 2 } = 112.61 , \quad \Sigma x y = 106.51 .$$
4. Explain why, if the underlying population has a bivariate Normal distribution, it would be preferable to carry out a hypothesis test based on the product moment correlation coefficient. Comment briefly on the significance of the product moment correlation coefficient in relation to that of Spearman’s rank correlation coefficient.

3 The number of calls received at an office per 5 minutes is modelled by a Poisson distribution with mean 3.2.

1. Find the probability of
   (A) exactly one call in a 5 -minute period,
   (B) at least 6 calls in a 5 -minute period.
2. Find the probability of
   (A) exactly one call in a 1 -minute period,
   (B) exactly one call in each of five successive 1-minute periods.
3. Use a suitable approximating distribution to find the probability of at most 45 calls in a period of 1 hour. Two assumptions required for a Poisson distribution to be a suitable model are that calls arrive
   • at a uniform average rate,
   • independently of each other.
   • Comment briefly on the validity of each of these assumptions if the office is
     (A) the enquiry department of a bank,
     (B) a police emergency control room.

4 The sexes and ages of a random sample of 300 runners taking part in marathons are classified as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between age group and sex. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
2. Does your analysis support the suggestion that women are less likely than men to enter marathons as they get older? Justify your answer. For marathons in general, on average \(3 \%\) of runners are 'Female, 50 and over'. The random variable \(X\) represents the number of 'Female, 50 and over' runners in a random sample of size 300.
3. Use a suitable approximating distribution to find \(\mathrm { P } ( X \geqslant 12 )\).