OCR S4 (Statistics 4) 2012 June

1 Independent random variables \(X\) and \(Y\) have distributions \(\mathrm { B } ( 7 , p )\) and \(\mathrm { B } ( 8 , p )\) respectively.

1. Explain why \(X + Y \sim \mathrm {~B} ( 15 , p )\).
2. Find \(\mathrm { P } ( X = 2 \mid X + Y = 5 )\).

2 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 4 x e ^ { - 2 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$

1. Show that the moment generating function ( mgf ) of \(X\) is $$\frac { 4 } { ( 2 - t ) ^ { 2 } } , \text { where } | t | < 2$$
2. Explain why the \(\operatorname { mgf }\) of \(- X\) is \(\frac { 4 } { ( 2 + t ) ^ { 2 } }\).
3. Two random observations of \(X\) are denoted by \(X _ { 1 }\) and \(X _ { 2 }\). What is the \(\operatorname { mgf }\) of \(X _ { 1 } - X _ { 2 }\) ?

3 Because of the large number of students enrolled for a university geography course and the limited accommodation in the lecture theatre, the department provides a filmed lecture. Students are randomly assigned to two groups, one to attend the lecture theatre and the other the film. At the end of term the two groups are given the same examination. The geography professor wishes to test whether there is a difference in the performance of the two groups and selects the marks of two random samples of students, 6 from the group attending the lecture theatre and 7 from the group attending the films. The marks for the two samples, ordered for convenience, are shown below.

1. Stating a necessary assumption, carry out a suitable non-parametric test, at the \(10 \%\) significance level, for a difference between the median marks of the two groups.
2. State conditions under which a two-sample \(t\)-test could have been used.
3. Assuming that the tests in parts (i) and (ii) are both valid, state with a reason which test would be preferable.

4 The random variable \(U\) has the distribution \(\operatorname { Geo } ( p )\).

1. Show, from the definition, that the probability generating function ( pgf ) of \(U\) is given by $$G _ { U } ( t ) = \frac { p t } { 1 - q t } , \text { for } | t | < \frac { 1 } { q } ,$$ where \(q = 1 - p\).
2. Explain why the condition \(| t | < \frac { 1 } { q }\) is necessary.
3. Use the pgf to obtain \(\mathrm { E } ( U )\). Each packet of Corn Crisp cereal contains a voucher and \(20 \%\) of the vouchers have a gold star. When 4 gold stars have been collected a gift can be claimed. Let \(X\) denote the number of packets bought by a family up to and including the one from which the \(4 ^ { \text {th } }\) gold star is obtained.
4. Obtain the pgf of \(X\).
5. Find \(\mathrm { P } ( X = 6 )\).

5 A one-tail sign test of a population median is to be carried out at the \(5 \%\) significance level using a sample of size \(n\).

1. Show by calculation that the test can never result in rejection of the null hypothesis when \(n = 4\). The coach of a college swimming team expects Elena, the best 50 m freestyle swimmer, to have a median time less than 30 seconds. Elena found from records of her previous 72 swims that 44 were less than 30 seconds and 28 were greater than 30 seconds.
2. Stating a necessary assumption, test at the \(5 \%\) significance level whether Elena's median time for the 50 m freestyle is less than 30 seconds.

6 The random variables \(S\) and \(T\) are independent and have joint probability distribution given in the table.

1. Show that \(a = 0.12\) and find the value of \(b\).
2. Find \(\mathrm { P } ( T - S = 1 )\).
3. Find \(\operatorname { Var } ( T - S )\).

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 1 + a x ) & - 2 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Show that \(| a | \leqslant \frac { 1 } { 2 }\).
2. Find \(\mathrm { E } ( X )\) in terms of \(a\).
3. Construct an unbiased estimator \(T _ { 1 }\) of \(a\) based on one observation \(X _ { 1 }\) of \(X\).
4. A second observation \(X _ { 2 }\) is taken. Show that \(T _ { 2 }\), where \(T _ { 2 } = \frac { 3 } { 8 } \left( X _ { 1 } + X _ { 2 } \right)\), is also an unbiased estimator of a.
5. Given that \(\operatorname { Var } ( X ) = \sigma ^ { 2 }\), determine which of \(T _ { 1 }\) and \(T _ { 2 }\) is the better estimator.

8 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.3\) and \(\mathrm { P } ( A \mid B ) = 0.6\).

1. Show that \(\mathrm { P } ( B ) \leqslant 0.5\).
2. Given also that \(\mathrm { P } ( A \cup B ) = x\), find \(\mathrm { P } ( B )\) in terms of \(x\).