OCR S4 (Statistics 4) 2016 June

1 Ten archers shot at targets with two types of bow. Their scores out of 100 are shown in the table.

1. Use the sign test, at the \(5 \%\) level of significance, to test the hypothesis that bow type \(P\) is better than bow type \(Q\).
2. Why would a Wilcoxon signed rank test, if valid, be a better test than the sign test?

2 Low density lipoprotein (LDL) cholesterol is known as 'bad' cholesterol.
15 randomly chosen patients, each with an LDL level of 190 mg per decilitre of blood, were given one of two treatments, chosen at random. After twelve weeks their LDL levels, in mg per decilitre, were as follows. Use a Wilcoxon rank sum test, at the \(5 \%\) level of significance, to test whether the LDL levels of patients given treatment \(B\) are lower than the LDL levels of patients given treatment \(A\).

3 The table shows the joint probability distribution of two random variables \(X\) and \(Y\).

1. Find \(\operatorname { Cov } ( X , Y )\).
2. Are \(X\) and \(Y\) independent? Give a reason for your answer.
3. Find \(\mathrm { P } ( X = 1 \mid X Y = 2 )\).

4 The continuous random variable \(Y\) has a uniform (rectangular) distribution on \([ a , b ]\), where \(a\) and \(b\) are constants.

1. Show that the moment generating function \(\mathrm { M } _ { Y } ( \mathrm { t } )\) of \(Y\) is \(\frac { \left( \mathrm { e } ^ { b t } - \mathrm { e } ^ { a t } \right) } { t ( b - a ) }\).
2. Use the series expansion of \(\mathrm { e } ^ { x }\) to show that the mean and variance of \(Y\) are \(\frac { 1 } { 2 } ( a + b )\) and \(\frac { 1 } { 12 } ( b - a ) ^ { 2 }\), respectively.

5 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = 0.75\).

1. Find \(\mathrm { P } ( A \cap B )\) and \(\mathrm { P } ( A \cup B )\).
2. Determine, giving a reason in each case,
   (a) whether \(A\) and \(B\) are mutually exclusive,
   (b) whether \(A\) and \(B\) are independent.
3. A further event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1\) and \(\mathrm { P } ( A \cap B \cap C ) = 0.05\). It is also given that \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C \right) = \mathrm { P } \left( A ^ { \prime } \cap B \cap C \right) = x\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 2 x\).
   Find \(\mathrm { P } ( C )\).

6 Andrew has five coins. Three of them are unbiased. The other two are biased such that the probability of obtaining a head when one of them is tossed is \(\frac { 3 } { 5 }\). Andrew tosses all five coins. It is given that the probability generating function of \(X\), the number of heads obtained on the unbiased coins, is \(\mathrm { G } _ { X } ( t )\), where $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 8 } + \frac { 3 } { 8 } t + \frac { 3 } { 8 } t ^ { 2 } + \frac { 1 } { 8 } t ^ { 3 }$$

1. Find \(G _ { Y } ( \mathrm { t } )\), the probability generating function of \(Y\), the number of heads on the biased coins.
2. The random variable \(Z\) is the total number of heads obtained when Andrew tosses all five coins. Find the probability generating function of \(Z\), giving your answer as a polynomial.
3. Find \(\mathrm { E } ( Z )\) and \(\operatorname { Var } ( Z )\).
4. Write down the value of \(\mathrm { P } ( Z = 3 )\).

7 A continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < a
1 - \frac { a ^ { 5 } } { y ^ { 5 } } & y \geqslant a \end{array} \right.$$ where \(a\) is a parameter.
Two independent observations of \(Y\) are denoted by \(Y _ { 1 }\) and \(Y _ { 2 }\). The smaller of them is denoted by S .

1. Show that \(P ( S > \mathrm { s } ) = \frac { a ^ { 10 } } { s ^ { 10 } }\) and hence find the probability density function of \(S\).
2. Show that \(S\) is not an unbiased estimator of \(a\), and construct an unbiased estimator of \(a , T _ { 1 }\) based on \(S\).
3. Construct another unbiased estimator of \(a , T _ { 2 }\), of the form \(k \left( Y _ { 1 } + Y _ { 2 } \right)\), where \(k\) is a constant to be found.
4. Without further calculation, explain how you would decide which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.