WJEC Further Unit 2 (Further Unit 2) 2023 June

1. The random variable \(X\) has mean 17 and variance 64 . The independent random variable \(Y\) has mean 10 and variance 16 . Find the value of
   1. \(\mathrm { E } ( 4 Y - 2 X + 1 )\),
   2. \(\quad \operatorname { Var } ( 4 Y - 5 X + 3 )\),
   3. \(\mathrm { E } \left( X ^ { 2 } Y \right)\).
   4. For a set of 30 pairs of observations of the variables \(x\) and \(y\), it is known that \(\sum x = 420\) and \(\sum y = 240\). The least squares regression line of \(y\) on \(x\) passes through the point with coordinates \(( 19,20 )\).
   5. Show that the equation of the regression line of \(y\) on \(x\) is \(y = 2 \cdot 4 x - 25 \cdot 6\) and use it to predict the value of \(y\) when \(x = 26\).
   6. State two reasons why your prediction in part (a) may not be reliable.
   7. It is known that the average lifetime of hair dryers from a certain manufacturer is 2 years. The lifetimes are exponentially distributed.
   8. Find the probability that the lifetime of a randomly selected hair dryer is between 1.8 and \(2 \cdot 5\) years.
   9. Given that \(20 \%\) of hair dryers have a lifetime of at least \(k\) years, find the value of \(k\).
   10. Jon buys his first hair dryer from the manufacturer today. He will replace his hair dryer with another from the same manufacturer immediately when it stops working. Find the probability that, in the next 5 years, Jon will have to replace more than 3 hair dryers.
   11. State one assumption that you have made in part (c).
   12. A continuous random variable \(X\) has cumulative distribution function \(F\) given by
   $$F ( x ) = \begin{cases} 0 & \text { for } x < 0
   \frac { 1 } { 4 } x & \text { for } 0 \leqslant x \leqslant 2
   \frac { 1 } { 480 } x ^ { 4 } + \frac { 7 } { 15 } & \text { for } 2 < x \leqslant b
   1 & \text { for } x > b \end{cases}$$
2. Show that \(b = 4\).
3. Find \(\mathrm { P } ( X \leqslant 2 \cdot 5 )\).
4. Write down the value of the lower quartile of \(X\).
5. Find the value of the upper quartile of \(X\).
6. Find, correct to three significant figures, the value of \(k\) that satisfies the equation \(\mathrm { P } ( X > 3 \cdot 5 ) = \mathrm { P } ( X < k )\).

5. (a) Give two circumstances where it may be more appropriate to use Spearman's rank correlation coefficient rather than Pearson's product moment correlation coefficient.
(b) A farmer needs a new tractor. The tractor salesman selects 6 tractors at random to show the farmer. The farmer ranks these tractors, in order of preference, according to their ability to meet his needs on the farm. The tractor salesman makes a note of the price and power take-off (PTO) of the tractors. Spearman's rank correlation coefficient between the farmer's ranks and the price is 0.9429 .

1. Test at the \(5 \%\) significance level whether there is an association between the price of a tractor and the farmer's judgement of the ability of the tractor to meet his needs on the farm.
2. Calculate Spearman's rank correlation coefficient between the farmer's rank and PTO.
3. How should the tractor salesman interpret the results in (i) and (ii)?

6. A company has 20 boats to hire out. Payment is always taken in advance and all 20 boats are hired out each day. A manager at the company notices that \(10 \%\) of groups do not turn up to take the boats, despite having already paid to hire them. The manager wishes to investigate whether the numbers of boats that do not get taken each day can be modelled by the binomial distribution \(B ( 20,0 \cdot 1 )\). The numbers of boats that were not taken for 110 randomly selected days are given below.

1. State suitable hypotheses to carry out a goodness of fit test.
2. Here is part of the table for a \(\chi ^ { 2 }\) goodness of fit test on the data.

   Number of boats not taken	0	1	2	3	4	5 or more
   Observed	10	35	29	25	8	3
   Expected	\(f\)	29.72	\(g\)	20.91	9.88	\(4 \cdot 75\)

   1. Calculate the values of \(f\) and \(g\).
   2. By completing the test, give the conclusion the manager should reach. The cost of hiring a boat is \(\pounds 15\). Since demand is high and the proportion of groups that do not turn up is also relatively high, the manager decides to take payment for 22 boats each day. She would give \(\pounds 20\) (a full refund and some compensation) to any group that has paid and turned up, but cannot take a boat out due to the overselling. Assume that the proportion of groups not turning up stays the same.
3. 1. Suggest a binomial model that the manager could use for the number of groups arriving expecting to hire a boat.
   2. Hence calculate the expected daily net income for the company following the manager's decision.
4. Is the manager justified in her decision? Give a reason for your answer.