OCR S4 (Statistics 4) 2007 June

1 For the events \(A\) and \(B , \mathrm { P } ( A ) = 0.3 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \right) = c\), where \(c \neq 0\).

1. Find \(\mathrm { P } ( A \cap B )\) in terms of \(c\).
2. Find \(\mathrm { P } ( B \mid A )\) and deduce that \(0.1 \leqslant c \leqslant 0.4\).

2 Of 9 randomly chosen students attending a lecture, 4 were found to be smokers and 5 were nonsmokers. During the lecture their pulse-rates were measured, with the following results in beats per minute. It may be assumed that these two groups of students were random samples from the student populations of smokers and non-smokers. Using a suitable Wilcoxon test at the \(10 \%\) significance level, test whether there is a difference in the median pulse-rates of the two populations.

3 The discrete random variables \(X\) and \(Y\) have the joint probability distribution given in the following table.

1. Show that \(\operatorname { Cov } ( X , Y ) = 0\).
2. Find the conditional distribution of \(X\) given that \(Y = 2\).

4 The levels of impurity in a particular alloy were measured using a random sample of 20 specimens. The results, in suitable units, were as follows.

1. Use the sign test, at the \(5 \%\) significance level, to decide if there is evidence that the population median level of impurity is greater than 2.70 .
2. State what other test might have been used, and give one advantage and one disadvantage this other test has over the sign test.

5 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { ( \alpha - 1 ) ! } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } & x \geqslant 0
0 & x < 0 \end{cases}$$ where \(\alpha\) is a positive integer.

1. Explain how you can deduce that \(\int _ { 0 } ^ { \infty } x ^ { \alpha - 1 } \mathrm { e } ^ { - x } \mathrm {~d} x = ( \alpha - 1 )\) !.
2. Write down an integral for the moment generating function \(\mathrm { M } _ { X } ( t )\) of \(X\) and show, by using the substitution \(x = \frac { u } { 1 - t }\), that \(\mathrm { M } _ { X } ( t ) = ( 1 - t ) ^ { - \alpha }\).
3. Use the moment generating function to find, in terms of \(\alpha\),
   (a) \(\mathrm { E } ( X )\),
   (b) \(\operatorname { Var } ( X )\).

6 The discrete random variable \(X\) takes the values 0 and 1 with \(\mathrm { P } ( X = 0 ) = q\) and \(\mathrm { P } ( X = 1 ) = p\), where \(p + q = 1\).

1. Write down the probability generating function of \(X\). The sum of \(n\) independent observations of \(X\) is denoted by \(S\).
2. Write down the probability generating function of \(S\), and name the distribution of \(S\).
3. Use the probability generating function of \(S\) to find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
4. The independent random variables \(Y\) and \(Z\) are such that \(Y\) has the distribution \(\mathrm { B } \left( 10 , \frac { 1 } { 2 } \right)\), and \(Z\) has probability generating function \(\mathrm { e } ^ { - ( 1 - t ) }\). Find the probability that the sum of one random observation of \(Y\) and one random observation of \(Z\) is equal to 2 .

7 The continuous random variable \(X\) has a uniform distribution over the interval \([ 0 , \theta ]\) so that the probability density function is given by $$f ( x ) = \begin{cases} \frac { 1 } { \theta } & 0 \leqslant x \leqslant \theta
0 & \text { otherwise } \end{cases}$$ where \(\theta\) is a positive constant. A sample of \(n\) independent observations of \(X\) is taken and the sample mean is denoted by \(\bar { X }\).

1. The estimator \(T _ { 1 }\) is defined by \(T _ { 1 } = 2 \bar { X }\). Show that \(T _ { 1 }\) is an unbiased estimator of \(\theta\). It is given that the probability density function of the largest value, \(U\), in the sample is $$g ( u ) = \begin{cases} \frac { n u ^ { n - 1 } } { \theta ^ { n } } & 0 \leqslant u \leqslant \theta
   0 & \text { otherwise } \end{cases}$$
2. Find \(\mathrm { E } ( U )\) and show that \(\operatorname { Var } ( U ) = \frac { n \theta ^ { 2 } } { ( n + 1 ) ^ { 2 } ( n + 2 ) }\).
3. The estimator \(T _ { 2 }\) is defined by \(T _ { 2 } = \frac { n + 1 } { n } U\). Given that \(T _ { 2 }\) is also an unbiased estimator of \(\theta\), show that \(T _ { 2 }\) is a more efficient estimator than \(T _ { 1 }\) for \(n > 1\).