OCR MEI Further Statistics Minor (Further Statistics Minor) Specimen

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 \(10\)th throw. [2]
2. Find the probability that she does not hit the bullseye in her first \(10\) throws. [1]
3. Write down the expected number of throws which it takes her to hit the bullseye for the first time. [1]

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.

\(x\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(P(X = x)\)	\(0.05\)	\(0.2\)	\(0.5\)	\(0.2\)	\(0.05\)

1. 1. Explain why \(\text{E}(X) = 2\). [1]
   2. Find \(\text{Var}(X)\). [3]
2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250X - 80\). Find

The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.

1. Explain why it would be inappropriate to test all the remote controls in this way. [1]
2. State an advantage of using random sampling in this context. [1]

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.

1. Let \(X\) be the number of points gained after shopping once. Find
2. Let \(Y\) be the number of points gained after shopping twice. Find
3. Find the probability of the most likely number of points gained after shopping twice. Justify your answer. [4]

1. State the conditions under which the Poisson distribution is an appropriate model for the number of emails received by one person in a day. [2]

Jane records the number of junk emails which she receives each day. During working hours (\(9\)am to \(5\)pm, Monday to Friday) the mean number of junk emails is \(7.4\) per day. Outside working hours (\(5\)pm to \(9\)am), 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".

1. Find the probability that the number of junk emails which she receives between \(9\)am and \(5\)pm on a Monday is
   1. exactly \(10\), [1]
   2. at least \(10\). [2]
2. 1. What assumption must you make to calculate the probability that the number of junk emails which she receives from \(9\)am Monday to \(9\)am Tuesday is at most \(20\)? [1]
   2. Find the probability. [2]

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.

Contestant	A	B	C	D	E	F	G
Age	66	51	39	29	9	22	14
Score	12	11	15	17	16	18	9

Table 5.1 \includegraphics{figure_1} Fig. 5.2 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. [1]
2. Calculate Spearman's rank correlation coefficient for the \(6\) remaining contestants. [3]
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. [5]
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. [1]

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.

Number of recaptures, \(x\)	0	1	2	3	4	5	6	7	8	9	10
Frequency	2	5	5	9	10	4	3	1	0	1	0

1. The sample mean of \(x\) is \(3.4\). Calculate the sample variance of \(x\). [2]
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. [1]

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. \includegraphics{figure_2}

1. State the null and alternative hypotheses for the test. [1]
2. Calculate the missing values in cells
3. Complete the test at the \(10\%\) significance level. [5]
4. 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. \includegraphics{figure_3} [3]

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]