OCR MEI Further Statistics Minor (Further Statistics Minor) Specimen

1 A darts player is trying to hit the bullseye on a dart board. On each throw the probability that she hits it is 0.05 , independently of any other throw.

1. Find the probability that she hits the bullseye for the first time on her 10th throw.
2. Find the probability that she does not hit the bullseye in her first 10 throws.
3. Write down the expected number of throws which it takes her to hit the bullseye for the first time.
4. Write down the expected number of throws which it takes her to hit the bullseye for the first time.

2 The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.

1. (A) Explain why \(\mathrm { E } ( X ) = 2\).
   (B) Find \(\operatorname { Var } ( X )\).
2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250 X - 80\).
   Find
   • \(\mathrm { E } ( Y )\) and
   • \(\operatorname { Var } ( Y )\).
   The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.
3. Explain why it would be inappropriate to test all the remote controls in this way.
4. State an advantage of using random sampling in this context. A website awards a random number of loyalty points each time a shopper buys from it. The shopper gets a whole number of points between 0 and 10 (inclusive). Each possibility is equally likely, each time the shopper buys from the website. Awards of points are independent of each other.
5. Let \(X\) be the number of points gained after shopping once. Find
   • the mean of \(X\)
   • the variance of \(X\).
   • Let \(Y\) be the number of points gained after shopping twice.
   Find
   • the mean of \(Y\)
   • the variance of \(Y\).
   • Find the probability of the most likely number of points gained after shopping twice. Justify your answer.
   • State the conditions under which the Poisson distribution is an appropriate model for the number of emails received by one person in a day.
   Jane records the number of junk emails which she receives each day. During working hours (9am to 5pm, Monday to Friday) the mean number of junk emails is 7.4 per day. Outside working hours ( 5 pm to 9am), the mean number of junk emails is 0.3 per hour. For the remainder of this question, you should assume that Poisson models are appropriate for the number of junk emails received during each of "working hours" and "outside working hours".
6. Find the probability that the number of junk emails which she receives between 9am and 5pm on a Monday is
   (A) exactly 10 ,
   (B) at least 10 .
7. (A) What assumption must you make to calculate the probability that the number of junk emails which she receives from 9am Monday to 9am Tuesday is at most 20?
   (B) Find the probability.

5 Each contestant in a talent competition is given a score out of 20 by a judge. The organisers suspect that the judge's scores are associated with the age of the contestant. Table 5.1 and the scatter diagram in Fig. 5.2 show the scores and ages of a random sample of 7 contestants. \begin{table}[h] \end{table} \begin{figure}[h] \end{figure} Contestant G did not finish her performance, so it is decided to remove her data.

1. Spearman's rank correlation coefficient between age and score, including all 7 contestants, is - 0.25 . Explain why Spearman's rank correlation coefficient becomes more negative when the data for contestant G is removed.
2. Calculate Spearman's rank correlation coefficient for the 6 remaining contestants.
3. Using this value of Spearman's rank correlation coefficient, carry out a hypothesis test at the \(5 \%\) level to investigate whether there is any association between age and score.
4. Briefly explain why it may be inappropriate to carry out a hypothesis test based on Pearson's product moment correlation coefficient using these data.

6 At a bird feeding station, birds are captured and ringed. If a bird is recaptured, the ring enables it to be identified. The table below shows the number of recaptures, \(x\), during a period of a month, for each bird of a particular species in a random sample of 40 birds.

1. The sample mean of \(x\) is 3.4. Calculate the sample variance of \(x\).
2. Briefly comment on whether the results of part (i) support a suggestion that a Poisson model might be a good fit to the data. The screenshot below shows part of a spreadsheet for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean of 3.4 has been used as an estimate of the Poisson parameter. Some values in the spreadsheet have been deliberately omitted.

   	A	B	C	D	E
   1	Number of recaptures	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   2	0 or 1	7	0.1468	5.8737	0.2160
   3	2	5			0.9560
   4	3	9	0.2186	8.7447	0.0075
   5	4	10	0.1858	7.4330	0.8865
   6	5	4	0.1264	5.0544
   7	\(\geq 6\)	5	0.1295	5.1783	0.0061

3. State the null and alternative hypotheses for the test.
4. Calculate the missing values in cells
   • C3,
   • D3 and
   • E6.
   • Complete the test at the \(10 \%\) significance level.
   • The screenshot below shows part of a spreadsheet for a \(\chi ^ { 2 }\) test for a different species of bird. Find the value of the Poisson parameter used.

   	A	B	C	D	E
   1	Number of recaptures	Observed frequency	Poisson probability	Expected frequency	Chi-squared contribution
   3	1	10	0.25716	12.8579	0.6352
   4	2	7	0.27002	13.5008	3.1302
   5	3	15	0.18901	9.4506	3.2587
   6	\(\geq 4\)	11	0.16136	8.0679	1.0656

7 A fair coin has + 1 written on the heads side and - 1 on the tails side. The coin is tossed 100 times. The sum of the numbers showing on the 100 tosses is the random variable \(Y\). Show that the variance of \(Y\) is 100 . [4] \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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