WJEC Further Unit 2 (Further Unit 2) 2018 June

1. The random variable \(X\) has the binomial distribution \(\mathrm { B } ( 12,0 \cdot 3 )\). The independent random variable \(Y\) has the Poisson distribution \(\mathrm { Po } ( 4 )\). Find
   1. \(\mathrm { E } ( X Y )\),
   2. \(\quad \operatorname { Var } ( X Y )\).
   3. The length of time a battery works, in tens of hours, is modelled by a random variable \(X\) with cumulative distribution function
   $$F ( x ) = \begin{cases} 0 & \text { for } x < 0
   \frac { x ^ { 3 } } { 432 } ( 8 - x ) & \text { for } 0 \leqslant x \leqslant 6
   1 & \text { for } x > 6 \end{cases}$$
2. Find \(P ( X > 5 )\).
3. A head torch uses three of these batteries. All three batteries must work for the torch to operate. Find the probability that the head torch will operate for more than 50 hours.
4. Show that the upper quartile of the distribution lies between \(4 \cdot 5\) and \(4 \cdot 6\).
5. Find \(f ( x )\), the probability density function for \(X\).
6. Find the mean lifetime of the batteries in hours.
7. The graph of \(f ( x )\) is given below.
   \includegraphics[max width=\textwidth, alt={}, center]{3cf25b54-d4a1-4d30-b632-3f6d3182a930-2_695_1463_1896_299}

3. A game at a school fete is played with a fair coin and a random number generator which generates random integers between 1 and 52 inclusive. It costs 50 pence to play the game. First, the player tosses the coin. If it lands on tails, the player loses. If it lands on heads, the player is allowed to generate a random number. If the number is 1 , the player wins \(\pounds 5\). If the number is between 2 and 13 inclusive, the player wins \(\pounds 1\). If the number is greater than 13 , the player loses.

1. Find the probability distribution of the player's profit.
2. Find the mean and standard deviation of the player's profit.
3. Given that 200 people play the game, calculate
   1. the expected number of players who win some money,
   2. the expected profit for the fete.

4. On a Welsh television game show, contestants are asked to guess the weights of a random sample of seven cows. The game show judges want to investigate whether there is positive correlation between the actual weights and the estimated weights. The results are shown below for one contestant.

1. Calculate Spearman's rank correlation coefficient for this data set.
2. Stating your hypotheses clearly, determine whether or not there is evidence at the \(5 \%\) significance level of a positive association between the actual weights and the weights as estimated by this contestant.
3. One of the game show judges says, "This contestant was good at guessing the weights of the cows." Comment on this statement.

5. A life insurance saleswoman investigates the number of policies she sells per day. The results for a random sample of 50 days are shown in the table below. She sees the same fixed number of clients each day. She would like to know whether the binomial distribution with parameters 6 and 0.6 is a suitable model for the number of policies she sells per day.

1. State suitable hypotheses for 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 policies sold	0	1	2	3	4	5	6
   Observed	2	2	9	12	15	9	1
   Expected	0.205	1.843	6.912	\(d\)	\(e\)	9.331	2.333

   1. Calculate the values of \(d\) and \(e\).
   2. Carry out the test using a 10\% level of significance and draw a conclusion in context.
3. What do the parameters 6 and 0.6 mean in this context?

6. A student, considering options for the future, collects data on education and salary. The table below shows the highest level of education attained and the salary bracket of a random sample of 664 people. By conducting a chi-squared test for independence, the student investigates the relationship between the highest level of education attained and the salary earned.

1. State the null and alternative hypotheses.
2. The table below shows the expected values. Calculate the value of \(k\).

   Expected values	Fewer than 5 GCSE	5 or more GCSE	3 A Levels	University degree	Post graduate qualification
   Less than £20000	\(k\)	24.23	26•19	35.46	10.57
   £20000 to £60000	49.40	103.67	112.02	151.68	\(45 \cdot 23\)
   More than £60000	10.05	21.09	\(22 \cdot 79\)	30.86	9.20

3. The following computer output is obtained. Calculate the values of \(m\) and \(n\).

   Chi Squared Contributions	Fewer than 5 GCSE	5 or more GCSE	3 A Levels	University degree	Post graduate qualification
   Less than £20000	3.604530799	\(m\)	1.46165	1.5686	0.03098
   £20000 to £60000	0.007272735	0.72535	4E-06	0.07264	0.50396
   More than £60000	\(4 \cdot 946619863\)	0.03897	169081	\(0 \cdot 55498\)	\(n\)

   X-squared \(= 19 \cdot 61301 , d f = 8 , p\)-value \(= 0 \cdot 0119\)
4. 1. Without carrying out any further calculations, explain how X-squared \(= 19 \cdot 61301\) (the \(\chi ^ { 2 }\) test statistic) was calculated.
   2. Comment on the values in the "Fewer than 5 GCSE" column of the table in part (c).
5. The student says that the highest levels of education lead to the highest paying jobs. Comment on the accuracy of the student's statement.

7. A university professor conducted some research into factors that affect job satisfaction. The four factors considered were Interesting work, Good wages, Job security and Appreciation of work done. The professor interviewed workers at 14 different companies and asked them to rate their companies on each of the factors. The workers' ratings were averaged to give each company a score out of 5 on each factor. Each company was also given a score out of 100 for Job satisfaction.
The following graph shows the part of the research concerning Job Satisfaction versus Interesting work. \begin{figure}[h] \end{figure}

1. Calculate the equation of the least squares regression line of Job satisfaction ( \(y\) ) on Interesting work ( \(x\) ), given the following summary statistics. $$\begin{array} { l l l } \sum x = 46 \cdot 2 , & \sum y = 898 , & S _ { x x } = 3 \cdot 48
   S _ { x y } = 49 \cdot 45 , & S _ { y y } = 1437 \cdot 714 , & n = 14 \end{array}$$
2. Give two reasons why it would be inappropriate for the professor to use this equation to calculate the score for Interesting work from a Job satisfaction score of 90.