OCR S4 (Statistics 4) 2018 June

1 A Wilcoxon signed-rank test is carried out at the \(5 \%\) level of significance on a random sample of size 32 . The hypotheses are \(\mathrm { H } _ { 0 } : m = m _ { 0 } , \mathrm { H } _ { 1 } : m < m _ { 0 }\) where \(m\) is the population median and \(m _ { 0 }\) is a specific numerical value. The value obtained for the test statistic \(T\) is 162 . Find the outcome of the test.

2 The distances from home to work, in km , of 8 men and 5 women were recorded and are given below. The workers were chosen at random. Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether there is a significant difference between the distances from home to work between men and women.

3 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.4\) and \(\mathrm { P } ( A \cup B ) = 0.8\).

1. Find \(\mathrm { P } ( A \cap B )\).
2. Find \(\mathrm { P } \left( A \cap B ^ { \prime } \right)\).
3. Find \(\mathrm { P } ( A \mid B )\). Events \(A\) and \(B\) are as above and a third event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1 , \mathrm { P } ( A \cap B \cap C ) = 0.05\), \(\mathrm { P } ( A \cap C ) = \mathrm { P } ( B \cap C )\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 3 \mathrm { P } \left( A ^ { \prime } \cap B \cap C ^ { \prime } \right)\).
4. Find \(\mathrm { P } ( C )\).

4 The random variable \(X\) has a \(\chi ^ { 2 }\) distribution with \(v\) degrees of freedom. The moment generating function of \(X\) is $$\mathrm { M } _ { X } ( t ) = ( 1 - 2 t ) ^ { - \frac { 1 } { 2 } v }$$

1. Show that \(\mathrm { E } ( X ) = v\).
2. Find \(\operatorname { Var } ( X )\).
3. Obtain the moment generating function of the sum \(Y\) of two independent \(\chi ^ { 2 }\) random variables, one with 6 degrees of freedom and the other with 8 degrees of freedom.
4. Identify the distribution of \(Y\).

5 The independent discrete random variables \(U\) and \(V\) can each take the values 1, 2 and 3, all with probability \(\frac { 1 } { 3 }\). The random variables \(X\) and \(Y\) are defined as follows: $$X = | U - V | , Y = U + V .$$

1. In the Printed Answer Book complete the table showing the joint probability distribution of \(X\) and \(Y\).
2. Find \(\operatorname { Cov } ( X , Y )\).
3. State with a reason whether \(X\) and \(Y\) are independent.
4. Find \(\mathrm { P } ( Y = 3 \mid X = 1 )\).

6 In each round of a quiz a contestant can answer up to three questions. Each correct answer scores 1 point and allows the contestant to go on to the next question. A wrong answer scores 0 points and the contestant is allowed no further question in that round. If all 3 questions are answered correctly 1 bonus point is scored, making a total score of 4 for the round. For a certain contestant, \(A\), the probability of giving a correct answer is \(\frac { 3 } { 4 }\), independently of any other question. The random variable \(X _ { r }\) is the number of points scored by \(A\) during the \(r ^ { \text {th } }\) round.

1. Find the probability generating function of \(X _ { r }\).
2. Use the probability generating function found in part (i) to find the mean and variance of \(X _ { r }\).
3. Write down an expression for the probability generating function of \(X _ { 1 } + X _ { 2 }\) and find the probability that \(A\) has a total score of 4 at the end of two rounds.

7 Two independent observations \(X _ { 1 }\) and \(X _ { 2 }\) are made of a continuous random variable with probability density function $$f ( x ) = \begin{cases} \frac { 1 } { \theta } & 0 \leqslant x \leqslant \theta
0 & \text { otherwise } \end{cases}$$ where \(\theta\) is a parameter whose value is to be estimated.

1. Find \(\mathrm { E } ( X )\).
2. Show that \(S _ { 1 } = X _ { 1 } + X _ { 2 }\) is an unbiased estimator of \(\theta\).
   \(L\) is the larger of \(X _ { 1 }\) and \(X _ { 2 }\), or their common value if they are equal.
3. Show that the probability density function of \(L\) is \(\frac { 2 l } { \theta ^ { 2 } }\) for \(0 \leqslant l \leqslant \theta\).
4. Find \(\mathrm { E } ( L )\).
5. Find an unbiased estimator \(S _ { 2 }\) of \(\theta\), based on \(L\).
6. Determine which of the two estimators \(S _ { 1 }\) and \(S _ { 2 }\) is the more efficient.