OCR S4 (Statistics 4) 2010 June

1 For the variables \(A\) and \(B\), it is given that \(\operatorname { Var } ( A ) = 9 , \operatorname { Var } ( B ) = 6\) and \(\operatorname { Var } ( 2 A - 3 B ) = 18\).

1. Find \(\operatorname { Cov } ( A , B )\).
2. State with a reason whether \(A\) and \(B\) are independent.

2 The probability generating function of the discrete random variable \(X\) is \(\frac { \mathrm { e } ^ { 4 t ^ { 2 } } } { \mathrm { e } ^ { 4 } }\). Find

1. \(\mathrm { E } ( X )\),
2. \(\mathrm { P } ( X = 2 )\).
   \(3 X _ { 1 }\) and \(X _ { 2 }\) are continuous random variables. Random samples of 5 observations of \(X _ { 1 }\) and 6 observations of \(X _ { 2 }\) are taken. No two observations are equal. The 11 observations are ranked, lowest first, and the sum of the ranks of the observations of \(X _ { 1 }\) is denoted by \(R\).

1. Assuming that all rankings are equally likely, show that \(\mathrm { P } ( R \leqslant 17 ) = \frac { 2 } { 231 }\). The marks of 5 randomly chosen students from School \(A\) and 6 randomly chosen students from School \(B\), who took the same examination, achieving different marks, were ranked. The rankings are shown in the table.

   Rank	1	2	3	4	5	6	7	8	9	10	11
   School	\(A\)	\(A\)	\(A\)	\(B\)	\(A\)	\(A\)	\(B\)	\(B\)	\(B\)	\(B\)	\(B\)

2. For a Wilcoxon rank-sum test, obtain the exact smallest significance level for which there is evidence of a difference in performance at the two schools.

4 The moment generating function of a continuous random variable \(Y\), which has a \(\chi ^ { 2 }\) distribution with \(n\) degrees of freedom, is \(( 1 - 2 t ) ^ { - \frac { 1 } { 2 } n }\), where \(0 \leqslant t < \frac { 1 } { 2 }\).

1. Find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\). For the case \(n = 1\), the sum of 60 independent observations of \(Y\) is denoted by \(S\).
2. Write down the moment generating function of \(S\) and hence identify the distribution of \(S\).
3. Use a normal approximation to estimate \(\mathrm { P } ( S \geqslant 70 )\).

5 In order to test whether the median salary of employees in a certain industry who had worked for three years was \(\pounds 19500\), the salaries \(x\), in thousands of pounds, of 50 randomly chosen employees were obtained.

1. The values \(| x - 19.5 |\) were calculated and ranked. No two values of \(x\) were identical and none was equal to 19.5 . The sum of the ranks corresponding to positive values of \(( x - 19.5 )\) was 867. Stating a required assumption, carry out a suitable test at the \(5 \%\) significance level.
2. If the assumption you stated in part (i) does not hold, what test could have been used?

6 Nuts and raisins occur in randomly chosen squares of a particular brand of chocolate. The numbers of nuts and raisins are denoted by \(N\) and \(R\) respectively and the joint probability distribution of \(N\) and \(R\) is given by $$f ( n , r ) = \begin{cases} c ( n + 2 r ) & n = 0,1,2 \text { and } r = 0,1,2
0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant.

1. Find the value of \(c\).
2. Find the probability that there is exactly one nut in a randomly chosen square.
3. Find the probability that the total number of nuts and raisins in a randomly chosen square is more than 2 .
4. For squares in which there are 2 raisins, find the mean number of nuts.
5. Determine whether \(N\) and \(R\) are independent.

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta
0 & \text { otherwise } \end{cases}$$ where \(\theta\) is an unknown positive constant.

1. Find \(\mathrm { E } \left( X ^ { n } \right)\), where \(n \neq - 2\), and hence write down the value of \(\mathrm { E } ( X )\).
2. Find
   (a) \(\operatorname { Var } ( X )\),
   (b) \(\operatorname { Var } \left( X ^ { 2 } \right)\).
3. Find \(\mathrm { E } \left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right)\) and \(\mathrm { E } \left( X _ { 1 } ^ { 2 } + X _ { 2 } ^ { 2 } + X _ { 3 } ^ { 2 } \right)\), where \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent observations of \(X\). Hence construct unbiased estimators, \(T _ { 1 }\) and \(T _ { 2 }\), of \(\theta\) and \(\operatorname { Var } ( X )\) respectively, which are based on \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\).
4. Find \(\operatorname { Var } \left( T _ { 2 } \right)\).

8 For the events \(L\) and \(M , \mathrm { P } ( L \mid M ) = 0.2 , \mathrm { P } ( M \mid L ) = 0.4\) and \(\mathrm { P } ( M ) = 0.6\).

1. Find \(\mathrm { P } ( L )\) and \(\mathrm { P } \left( L ^ { \prime } \cup M ^ { \prime } \right)\).
2. Given that, for the event \(N , \mathrm { P } ( N \mid ( L \cap M ) ) = 0.3\), find \(\mathrm { P } \left( L ^ { \prime } \cup M ^ { \prime } \cup N ^ { \prime } \right)\).