WJEC Further Unit 2 (Further Unit 2) 2019 June

1. (a) Sketch a scatter diagram of a dataset for which Spearman's rank correlation coefficient is + 1 , but the product moment correlation coefficient is less than 1 .

Two judges were judging cheese at the UK Cheese Festival. There were 8 blue cheeses in a particular category. The rankings are shown below. (b) Calculate Spearman's rank correlation coefficient for this dataset.
(c) By sketching a scatter diagram of the rankings, or otherwise, comment on the extent to which the judges agree.

2. The probability of winning a certain game at a funfair is \(p\). Aman plays the game 5 times and Boaz plays the game 8 times. The independent random variables \(X\) and \(Y\) denote the number of wins for Aman and Boaz respectively.

1. Given that \(\mathrm { E } ( X Y ) = 6 \cdot 4\), calculate \(p\).
2. Find \(\operatorname { Var } ( X Y )\).

3. The number of claims made to the home insurance department of an insurance company follows a Poisson distribution with mean 4 per day.

1. Find the probability that more than 11 claims are made in a 2 -day period. The number of claims made in a day to the pet insurance department of the same company follows a Poisson distribution with parameter \(\lambda\). An insurance company worker notices that the probability of two claims being made in a day is three times the probability of four claims being made in a day.
2. Determine the value of \(\lambda\). The car insurance department models the length of time between claims for drivers aged 17 to 21 as an exponential distribution with mean 10 months. Rachel is 17 years old and has just passed her test. Her father says he will give her the car that they share if she does not make a claim in the first 12 months.
3. What is the probability that her father gives her the car?

4. The continuous random variable, \(X\), has the following probability density function $$f ( x ) = \begin{cases} k x & \text { for } 0 \leqslant x < 1
k x ^ { 3 } & \text { for } 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
\includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-3_851_935_826_678}

1. Show that \(k = \frac { 4 } { 17 }\).
2. Determine \(\mathrm { E } ( X )\).
3. Calculate \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).

5. Chris is investigating the distribution of birth months for ice hockey players. He collects data for 869 randomly chosen National Hockey League (NHL) players. He decides to carry out a chi-squared test. Using a spreadsheet, he produces the following output.

1. By considering the output, state the null hypothesis that Chris is testing. State what conclusion Chris should reach and explain why. Chris now wonders if Premier League football players' birth months are distributed uniformly throughout the year. He collects the birth months of 75 randomly selected Premier League footballers. This information is shown in the table below.

   Jan	Feb	Mar	Apr	May	Jun	Jul	Aug	Sep	Oct	Nov	Dec
   3	7	11	4	12	2	6	6	5	8	5	6

2. Carry out the chi-squared goodness of fit test at the 10\% significance level that Chris should use to conduct his investigation.

6. The University of Arizona surveyed a large number of households. One purpose of the survey was to determine if annual household income could be predicted from size of family home. The graph of Annual household income, \(y\), versus Size of family home, \(x\), is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-5_616_1257_566_365}

1. State the limitations of using the regression line above with reference to the scatter diagram. The data for size of family homes between 2000 and 3000 square feet are shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{4ecf99c5-c4b3-41b7-a8df-a7c2ca7fcd6a-5_652_1244_1516_360} Summary statistics for these data are as follows. $$\begin{array} { r c c } \sum x = 93160 & \sum y = 3907142 & n = 37
   S _ { x x } = 2869673.03 & S _ { y y } = 44312797167 & S _ { x y } = 348512820 \cdot 6 \end{array}$$
2. Calculate the equation of the least squares regression line to predict Annual household income from Size of family home for these data.

7. An article published in a medical journal investigated sports injuries in adolescents' ball games: football, handball and basketball. In a study of 906 randomly selected adolescent players in the three ball games, 379 players incurred injuries over the course of one year of playing the sport. Rhian wants to test whether there is an association between the site of injury and the sport played. A summary of the injuries is shown in the table below.

1. Calculate the values of \(A , B , C\) in the tables below.

   \multirow{2}{*}{}		Site of injury
   	Expected values	Shoulder/ Arm	Hand/ Fingers	Thigh/ Leg	Knee	Ankle	Foot	Other
   \multirow{3}{*}{sodod}	Football	13.1029	28.7256	27.7177	27.7177	57.9551	21.6702	14.1108
   	Handball	7.8892	17.2955	16.6887	16.6887	A	13.0475	8.4960
   	Basketball	5.0079	10.9789	10.5937	10.5937	22.1504	8.2823	5.3931

   \multirow{2}{*}{}	\multirow[b]{2}{*}{Chi-Squared Contributions}	Site of injury
   		Shoulder/ Arm	Hand/ Fingers	Thigh/ Leg	Knee	Ankle	Foot	Other
   \multirow{3}{*}{sodoct}	Football	1.98732	23.03890	\(10 \cdot 77575\)	2.47484	\(B\)	9.47586	0.31575
   	Handball	4.73333	4.38079	C	0.17087	1.44690	3.80664	0.73331
   	Basketball	0.20286	26.38865	4.10400	4.10400	0.00102	6.40306	3.93521

2. Given that the test statistic, \(X ^ { 2 }\), is 116.16, carry out the significance test at the \(5 \%\) level.
3. Which site of injury most affects the conclusion of this test? Comment on your answer. Rhian also analyses the data on the type of contact that caused the injuries and the sport in which they occur, shown in the table below.

   Observed values	Ball	Opponent	Surface	None	Total
   Football	17	68	17	92	194
   Handball	23	34	19	38	114
   Basketball	28	17	12	14	71
   Total	68	119	48	144	379

   The chi-squared test statistic is 46.0937 . Rhian notes that this value is smaller than 116.16 , the test statistic in part (b). She concludes that there is weaker evidence for association in this case than there was in part (b).
4. State Rhian's misconception and explain what she should consider instead. \section*{END OF PAPER}