OCR S4 (Statistics 4) 2014 June

1 A teacher believes that the calculator paper in a GCSE Mathematics examination was easier than the non-calculator paper. The marks of a random sample of ten students are shown in the table.

1. Use a Wilcoxon signed-rank test, at the \(5 \%\) significance level, to test the teacher's belief.
2. State the assumption necessary for this test to be applied.

2 During an outbreak of a disease, it is known that \(68 \%\) of people do not have the disease. Of people with the disease, \(96 \%\) react positively to a test for diagnosing it, as do \(m \%\) of people who do not have the disease.

1. In the case \(m = 8\), find the probability that a randomly chosen person has the disease, given that the person reacts positively to the test.
2. What value of \(m\) would be required for the answer to part (i) to be 0.95 ?

3 The discrete random variable \(X\) has probability generating function \(\frac { t } { a - b t }\), where \(a\) and \(b\) are constants.

1. Find a relationship between \(a\) and \(b\).
2. Use the probability generating function to find \(\mathrm { E } ( X )\) in terms of \(a\), giving your answer as simply as possible.
3. Expand the probability generating function as a power series, as far as the term in \(t ^ { 3 }\), giving the coefficients in terms of \(a\) and \(b\).
4. Name the distribution for which \(\frac { t } { a - b t }\) is the probability generating function, and state its parameter(s) in terms of \(a\).

4 The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } x & 0 \leqslant x \leqslant 1
2 - x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{array} \right.$$

1. Show that the moment generating function of \(X\) is \(\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }\).
   \(Y _ { 1 }\) and \(Y _ { 2 }\) are independent observations of a random variable \(Y\). The moment generating function of \(Y _ { 1 } + Y _ { 2 }\) is \(\frac { \left( \mathrm { e } ^ { t } - 1 \right) ^ { 2 } } { t ^ { 2 } }\).
2. Write down the moment generating function of \(Y\).
3. Use the expansion of \(\mathrm { e } ^ { t }\) to find \(\operatorname { Var } ( Y )\).
4. Deduce the value of \(\operatorname { Var } ( X )\).

5 Two discrete random variables \(X\) and \(Y\) have a joint probability distribution defined by $$\mathrm { P } ( X = x , Y = y ) = a ( x + y + 1 ) \quad \text { for } x = 0,1,2 \text { and } y = 0,1,2 ,$$ where \(a\) is a constant.

1. Show that \(a = \frac { 1 } { 27 }\).
2. Find \(\mathrm { E } ( X )\).
3. Find \(\operatorname { Cov } ( X , Y )\).
4. Are \(X\) and \(Y\) independent? Give a reason for your answer.
5. Find \(\mathrm { P } ( X = 1 \mid Y = 2 )\).

6 A Wilcoxon rank-sum test with samples of sizes 11 and 12 is carried out.

1. What is the least possible value of the test statistic \(W\) ?
2. The null hypothesis is that the two samples came from identical populations. Given that the null hypothesis was rejected at the \(1 \%\) level using a 2 -tail test, find the set of possible values of \(W\).

7 The continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c l } \frac { k } { ( x + \theta ) ^ { 5 } } & \text { for } x \geqslant 0
0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant and \(\theta\) is a parameter taking positive values.

1. Find an expression for \(k\) in terms of \(\theta\).
2. Show that \(\mathrm { E } ( X ) = \frac { 1 } { 3 } \theta\). You are given that \(\operatorname { Var } ( X ) = \frac { 2 } { 9 } \theta ^ { 2 }\). A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) of \(n\) observations of \(X\) is obtained. The estimator \(T _ { 1 }\) is defined as \(T _ { 1 } = \frac { 3 } { n } \sum _ { i = 1 } ^ { n } X _ { i }\).
3. Show that \(T _ { 1 }\) is an unbiased estimator of \(\theta\), and find the variance of \(T _ { 1 }\).
4. A second unbiased estimator \(T _ { 2 }\) is defined by \(T _ { 2 } = \frac { 1 } { 3 } \left( X _ { 1 } + 3 X _ { 2 } + 5 X _ { 3 } \right)\). For the case \(n = 3\), which of \(T _ { 1 }\) and \(T _ { 2 }\) is more efficient? \section*{OCR}