OCR S4 (Statistics 4) 2017 June

1 A meteorologist claims that the median daily rainfall in London is 2.2 mm . A single sample sign test is to be used to test the claim, using the following hypotheses:
\(\mathrm { H } _ { 0 }\) : a sample comes from a population with median 2.2,
\(\mathrm { H } _ { 1 }\) : the sample does not come from a population with median 2.2.
30 randomly selected observations of daily rainfall in London are compared with 2.2, and given a '+' sign if greater than 2.2 and a '-' sign if less than 2.2. (You may assume that no data values are exactly equal to 2.2.) The test is to be carried out at the \(5 \%\) level of significance. Let the number of ' + ' signs be \(k\). Find, in terms of \(k\), the critical region for the test showing the values of any relevant probabilities.

2 The independent discrete random variables \(X\) and \(Y\) can take the values 0,1 and 2 with probabilities as given in the tables. \(\quad\)

\(y\)	0	1	2
\(\mathrm { P } ( Y = y )\)	0.5	0.3	0.2

The random variables \(U\) and \(V\) are defined as follows: $$U = X Y , V = | X - Y | .$$

1. In the Printed Answer Book complete the table giving the joint distribution of \(U\) and \(V\).
2. Find \(\operatorname { Cov } ( U , V )\).
3. Find \(\mathrm { P } ( U V = 0 \mid V = 2 )\).

3 For events \(A , B\) and \(C\) it is given that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.5 , \mathrm { P } ( C ) = 0.4\) and \(\mathrm { P } ( A \cap B \cap C ) = 0.1\). It is also given that events \(A\) and \(B\) are independent and that events \(A\) and \(C\) are independent.

1. Find \(\mathrm { P } ( B \mid A )\).
2. Given also that events \(B\) and \(C\) are independent, find \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \cap C ^ { \prime } \right)\).
3. Given instead that events \(B\) and \(C\) are not independent, find the greatest and least possible values of \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \cap C ^ { \prime } \right)\).

4 The heights of eleven randomly selected primary school children are measured. The results, in metres, are

1. Use a Wilcoxon rank-sum test, at the \(1 \%\) significance level, to test whether primary school girls are taller than primary school boys.
2. It is decided to repeat the test, using larger random samples. The heights of twenty girls and eighteen boys are measured. Find the greatest value of the test statistic \(W\) which will result in the conclusion that there is evidence, at the \(1 \%\) level of significance, that primary school girls are taller than primary school boys.

5 The discrete random variable \(X\) is such that \(\mathrm { P } ( X = x ) = \frac { 3 } { 4 } \left( \frac { 1 } { 4 } \right) ^ { x } , x = 0,1,2 , \ldots\).

1. Show that the moment generating function of \(X , \mathrm { M } _ { X } ( t )\), can be written as \(\mathrm { M } _ { X } ( t ) = \frac { 3 } { 4 - \mathrm { e } ^ { t } }\).
2. Find the range of values of \(t\) for which the formula for \(\mathrm { M } _ { X } ( t )\) in part (i) is valid.
3. Use \(\mathrm { M } _ { X } ( t )\) to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

6 The continuous random variable \(Z\) has probability density function $$f ( z ) = \left\{ \begin{array} { c c } \frac { 4 z ^ { 3 } } { k ^ { 4 } } & 0 \leqslant z \leqslant k
0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a parameter whose value is to be estimated.

1. Show that \(\frac { 5 Z } { 4 }\) is an unbiased estimator of \(k\).
2. Find the variance of \(\frac { 5 Z } { 4 }\). The parameter \(k\) can also be estimated by making observations of a random variable \(X\) which has mean \(\frac { 1 } { 2 } k\) and variance \(\frac { 1 } { 12 } k ^ { 2 }\). Let \(Y = X _ { 1 } + X _ { 2 } + X _ { 3 }\) where \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent observations of \(X\).
3. \(c Y\) is also an unbiased estimator of \(k\). Find the value of \(c\).
4. For the value of \(c\) found in part (iii), determine which of \(\frac { 5 Z } { 4 }\) and \(c Y\) is the more efficient estimator of \(k\).

7 The discrete random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t ) = \frac { 1 } { 126 } t \left( 64 - t ^ { 6 } \right) \left( 1 - \frac { t } { 2 } \right) ^ { - 1 }\).

1. Find \(\mathrm { P } ( Y = 3 )\).
2. Find \(\mathrm { E } ( Y )\).