OCR MEI S2 (Statistics 2) 2012 January

1 Nine long-distance runners are starting an exercise programme to improve their strength. During the first session, each of them has to do a 100 metre run and to do as many push-ups as possible in one minute. The times taken for the run, together with the number of push-ups each runner achieves, are shown in the table.

1. Draw a scatter diagram to illustrate the data.
2. Calculate the value of Spearman's rank correlation coefficient.
3. Carry out a hypothesis test at the \(5 \%\) significance level to examine whether there is any association between time taken for the run and number of push-ups achieved.
4. Under what circumstances is it appropriate to carry out a hypothesis test based on the product moment correlation coefficient? State, with a reason, which test is more appropriate for these data.

2 The number of printing errors per page in a book is modelled by a Poisson distribution with a mean of 0.85 .

1. State conditions for a Poisson distribution to be a suitable model for the number of printing errors per page.
2. A page is chosen at random. Find the probability of
   (A) exactly 1 error on this page,
   (B) at least 2 errors on this page. 10 pages are chosen at random.
3. Find the probability of exactly 10 errors in these 10 pages.
4. Find the least integer \(k\) such that the probability of there being \(k\) or more errors in these 10 pages is less than \(1 \%\). 30 pages are chosen at random.
5. Use a suitable approximating distribution to find the probability of no more than 30 errors in these 30 pages.

3 The lifetime of a particular type of light bulb is \(X\) hours, where \(X\) is Normally distributed with mean 1100 and variance 2000.

1. Find \(\mathrm { P } ( 1100 < X < 1200 )\).
2. Use a suitable approximating distribution to find the probability that, in a random sample of 100 of these light bulbs, no more than 40 have a lifetime between 1100 and 1200 hours.
3. A factory has a large number of these light bulbs installed. As soon as \(1 \%\) of the bulbs have come to the end of their lifetimes, it is company policy to replace all of the bulbs. After how many hours should the bulbs need to be replaced?
4. The bulbs are to be replaced by low-energy bulbs. The lifetime of these bulbs is Normally distributed and the mean is claimed by the manufacturer to be 7000 hours. The standard deviation is known to be 100 hours. A random sample of 25 low-energy bulbs is selected. Their mean lifetime is found to be 6972 hours. Carry out a 2 -tail test at the \(10 \%\) level to investigate the claim.
   [0pt] [Question 4 is printed overleaf.]

4 Birds are observed at feeding stations in three different places - woodland, farm and garden. The numbers of finches, thrushes and tits observed at each site are summarised in the table. The birds observed are regarded as a random sample from the population of birds of these species that use these feeding stations. The expected frequencies under the null hypothesis for the usual \(\chi ^ { 2 }\) test are shown in the table below.

1. Verify that the entry 9.3180 is correct. The corresponding contributions to the test statistic are shown in the table below.

   \multirow{2}{*}{Contribution}		Place
   		Farm	Garden	Woodland
   \multirow{3}{*}{Species}	Thrushes	11.6539	60.7489	19.2192
   	Tits	2.0017	20.5201	9.5969
   	Finches	6.3332	6.1108	0.0000

2. Verify that the entry 6.3332 is correct.
3. Carry out the test at the \(1 \%\) level of significance.
4. For each place, use the table of contributions to comment briefly on the differences between the observed and expected distributions of species.