OCR Further Statistics AS (Further Statistics AS) 2024 June

1 The random variable \(W\) can take values 1,2 or 3 and has a discrete uniform distribution.

1. Write down the value of \(\mathrm { E } ( 2 W )\).
2. Find the value of \(\operatorname { Var } ( 2 W )\).
3. Determine the value of the constant \(k\) for which \(\mathrm { E } ( 2 \mathrm {~W} + \mathrm { k } ) = \operatorname { Var } ( 2 \mathrm {~W} + \mathrm { k } )\). The random variable \(S\) has the probability distribution shown in the following table.

   \(S\)	2	3	4	5	6
   \(P ( S = S )\)	\(\frac { 2 } { 9 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 1 } { 3 }\)	\(\frac { 1 } { 9 }\)	\(\frac { 2 } { 9 }\)

4. Calculate \(\operatorname { Var } ( S )\).

2 For a random sample of 160 employees of a large company, the principal method of transport for getting to work, arranged according to grade of employee, is shown in the table. A test is carried out at the \(5 \%\) significance level of whether there is association between grade of employee and method of transport.

1. State appropriate hypotheses for the test. The contributions to the test statistic are shown in the following table, correct to 3 decimal places.

   Grade	Walk or cycle	Private motorised transport	Public transport
   A	1.157	0.289	1.929
   B	1.878	0.225	0.327
   C	2.006	1.800	0.083

2. Show how the value 0.225 is obtained.
3. Complete the test, stating the conclusion.
4. Which combination of grade of employee and method of transport most strongly suggests association? Justify your answer.

3 The ages, \(x\) years, and the reaction time, \(t\) seconds, in an experiment carried out on a sample of 15 volunteers are summarised as follows.
\(n = 15 \quad \sum x = 762 \quad \sum t = 8.7 \quad \sum x ^ { 2 } = 44204 \quad \sum t ^ { 2 } = 5.65 \quad \sum x t = 490.1\)

1. Calculate the value of the product moment correlation coefficient between \(x\) and \(t\).
2. Calculate the equation of the line of regression of \(t\) on \(x\). Give your answer in the form \(\mathrm { t } = \mathrm { a } + \mathrm { bx }\) where \(a\) and \(b\) are constants to be determined.
3. Explain the relevance of the quantity \(\sum ( t - a - b x ) ^ { 2 }\) to your answer to part (b).
4. Estimate the reaction time, in seconds, for a volunteer aged 42. It is subsequently decided to measure the reaction time in tenths of a second rather than in seconds (so, for example, a time of 0.6 seconds would now be recorded as 6 ).
5. 1. State what effect, if any, this change would have on your answer to part (a).
   2. State what effect, if any, this change would have on your answer to part (b). It is known that the sample of 15 volunteers consisted almost entirely of students and retired people.
6. Using this information, and the value of the product moment correlation coefficient, comment on the reliability of your estimate in part (d).

1. Find the probability that 4 telephone calls are received in a randomly chosen one-minute period.
2. A sample of 10 independent observations of \(X\) is obtained. Find the expected number of these 10 observations that are in the interval \(2 < X < 8\). It is also known that
   \(P ( X + Y = 4 ) = \frac { 27 } { 8 } P ( X = 2 ) \times P ( Y = 2 )\).
3. Determine the possible values of \(\mathrm { E } ( Y )\).
4. Explain where in your solution to part (c) you have used the assumption that telephone calls and e-mails are received independently of one another.

5 In a fashion competition, two judges gave marks to a large number of contestants. The value of Spearman's rank correlation coefficient, \(\mathrm { r } _ { \mathrm { s } }\), between the marks given to 7 randomly chosen contestants is \(\frac { 27 } { 28 }\).

1. An excerpt from the table of critical values of \(\mathrm { r } _ { \mathrm { s } }\) is shown below. \section*{Critical values of Spearman's rank correlation coefficient}

   	1-tail test	5\%	2.5\%	1\%	0.5\%
   	2-tail test	10\%	5\%	2\%	1\%
   \multirow{3}{*}{\(n\)}	6	0.8286	0.8857	0.9429	1.0000
   	7	0.7143	0.7857	0.8929	0.9286
   	8	0.6429	0.7381	0.8333	0.8810

   Test whether there is evidence, at the 1\% significance level, that the judges agree with each another. The marks given by the two judges to the 7 randomly chosen contestants were as follows, where \(x\) is an integer.

   Contestant	A	B	C	D	\(E\)	\(F\)	G
   Judge 1	64	65	67	78	79	80	86
   Judge 2	61	63	78	80	81	90	\(x\)

2. Use the value \(\mathrm { r } _ { \mathrm { s } } = \frac { 27 } { 28 }\) to determine the range of possible values of \(x\).
3. Give a reason why it might be preferable to use the product moment correlation coefficient rather than Spearman's rank correlation coefficient in this context.

6 Anika walks along a street that contains parked cars. The number of cars that Anika passes, up to and including the first car that is white, is denoted by \(X\).

1. State two assumptions needed for \(X\) to be well modelled by a geometric distribution. Assume now that \(X\) can be well modelled by the distribution \(\operatorname { Geo } ( p )\), where \(0 < p < 1\).
2. For \(p = 0.1\), find \(\mathrm { P } ( X > 6 )\). The number of cars that Anika passes, up to but not including the first car that is white, is denoted by \(Y\).
3. For a general value of \(p\), determine a simplified expression for \(\mathrm { E } ( Y ) \div \operatorname { Var } ( Y )\), in terms of \(p\). Ben walks along a different street that also contains parked cars. The number of cars that Ben passes, up to and including the first white car on which the last digit of the number plate is even is denoted by \(Z\). It may be assumed that \(Z\) can be well modelled by the distribution \(\operatorname { Geo } \left( \frac { 1 } { 2 } p \right)\), where \(p\) is the parameter of the distribution of \(X\). It is given that \(\mathrm { P } ( \mathrm { Z } = 3 ) = \mathrm { kP } ( \mathrm { X } = 3 )\), where \(k\) is a positive constant.
4. Determine the range of possible values of \(k\).