OCR S4 (Statistics 4) 2009 June

2 A company wishes to buy a new lathe for making chair legs. Two models of lathe, 'Allegro' and 'Vivace', were trialled. The company asked 12 randomly selected employees to make a particular type of chair leg on each machine. The times, in seconds, for each employee are shown in the table. The company wishes to test whether there is any difference in average times for the two machines.

1. State the circumstances under which a non-parametric test should be used.
2. Use two different non-parametric tests and show that they lead to different conclusions at the 5\% significance level.
3. State, with a reason, which conclusion is to be preferred.

3 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \mathrm { e } ^ { 2 x } & x < 0
\mathrm { e } ^ { - 2 x } & x \geqslant 0 \end{cases}$$

1. Show that the moment generating function of \(X\) is \(\frac { 4 } { 4 - t ^ { 2 } }\), where \(| t | < 2\), and explain why the condition \(| t | < 2\) is necessary.
2. Find \(\operatorname { Var } ( X )\).

4 The probability generating function of the discrete random variable \(Y\) is given by $$\mathrm { G } _ { Y } ( t ) = \frac { a + b t ^ { 3 } } { t }$$ where \(a\) and \(b\) are constants.

1. Given that \(\mathrm { E } ( Y ) = - 0.7\), find the values of \(a\) and \(b\).
2. Find \(\operatorname { Var } ( Y )\).
3. Find the probability that the sum of 10 random observations of \(Y\) is - 7 .

5 Alana and Ben work for an estate agent. The joint probability distribution of the number of houses they sell in a randomly chosen week, \(X _ { A }\) and \(X _ { B }\) respectively, is shown in the table.
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1. Find \(\mathrm { E } \left( X _ { A } \right)\) and \(\operatorname { Var } \left( X _ { A } \right)\).
2. Determine whether \(X _ { A }\) and \(X _ { B }\) are independent.
3. Given that \(\mathrm { E } \left( X _ { B } \right) = 1.15 , \operatorname { Var } \left( X _ { B } \right) = 0.8275\) and \(\mathrm { E } \left( X _ { A } X _ { B } \right) = 1.09\), find \(\operatorname { Cov } \left( X _ { A } , X _ { B } \right)\) and \(\operatorname { Var } \left( X _ { A } - X _ { B } \right)\).
4. During a particular week only one house was sold by Alana and Ben. Find the probability that it was sold by Alana.

6 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < a ,
\mathrm { e } ^ { - ( x - a ) } & x \geqslant a , \end{cases}$$ where \(a\) is a constant. \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) are \(n\) independent observations of \(X\), where \(n \geqslant 4\).

1. Show that \(\mathrm { E } ( X ) = a + 1\).
   \(T _ { 1 }\) and \(T _ { 2 }\) are proposed estimators of \(a\), where $$T _ { 1 } = X _ { 1 } + 2 X _ { 2 } - X _ { 3 } - X _ { 4 } - 1 \quad \text { and } \quad T _ { 2 } = \frac { X _ { 1 } + X _ { 2 } } { 4 } + \frac { X _ { 3 } + X _ { 4 } + \ldots + X _ { n } } { 2 ( n - 2 ) } - 1 .$$
2. Show that \(T _ { 1 }\) and \(T _ { 2 }\) are unbiased estimators of \(a\).
3. Determine which is the more efficient estimator.
4. Suggest another unbiased estimator of \(a\) using all of the \(n\) observations.

7 A particular disease occurs in a proportion \(p\) of the population of a town. A diagnostic test has been developed, in which a positive result indicates the presence of the disease. It has a probability 0.98 of giving a true positive result, i.e. of indicating the presence of the disease when it is actually present. The test will give a false positive result with probability 0.08 when the disease is not present. A randomly chosen person is given the test.

1. Find, in terms of \(p\), the probability that
   (a) the person has the disease when the result is positive,
   (b) the test will lead to a wrong conclusion. It is decided that if the result of the test on someone is positive, that person is tested again. The result of the second test is independent of the result of the first test.
2. Find the probability that the person has the disease when the result of the second test is positive.
3. The town has 24000 children and plans to test all of them at a cost of \(\pounds 5\) per test. Assuming that \(p = 0.001\), calculate the expected total cost of carrying out these tests.