OCR S3 (Statistics 3) 2009 January

1 At a particular hospital, admissions of patients as a result of visits to the Accident and Emergency Department occur randomly at a uniform average rate of 0.75 per day. Independently, admissions that result from G.P. referrals occur randomly at a uniform average rate of 6.4 per week. The total number of admissions from these two causes over a randomly chosen period of four weeks is denoted by \(T\). State the distribution of \(T\) and obtain its expectation and variance.

2 The continuous random variable \(U\) has (cumulative) distribution function given by $$\mathrm { F } ( u ) = \begin{cases} \frac { 1 } { 5 } \mathrm { e } ^ { u } & u < 0
1 - \frac { 4 } { 5 } \mathrm { e } ^ { - \frac { 1 } { 4 } u } & u \geqslant 0 \end{cases}$$

1. Find the upper quartile of \(U\).
2. Find the probability density function of \(U\).

3 In a random sample of credit card holders, it was found that \(28 \%\) of them used their card for internet purchases.

1. Given that the sample size is 1200 , find a \(98 \%\) confidence interval for the percentage of all credit card holders who use their card for internet purchases.
2. Estimate the smallest sample size for which a \(98 \%\) confidence interval would have a width of at most \(5 \%\), and state why the value found is only an estimate.

4 The weekly sales of petrol, \(X\) thousand litres, at a garage may be modelled by a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} c & 25 \leqslant x \leqslant 45
0 & \text { otherwise } \end{cases}$$ where \(c\) is a constant. The weekly profit, in \(\pounds\), is given by \(( 400 \sqrt { X } - 240 )\).

1. Obtain the value of \(c\).
2. Find the expected weekly profit.
3. Find the probability that the weekly profit exceeds \(\pounds 2000\).

5 The concentration level of mercury in a large lake is known to have a normal distribution with standard deviation 0.24 in suitable units. At the beginning of June 2008, the mercury level was measured at five randomly chosen places on the lake, and the sample mean is denoted by \(\bar { x } _ { 1 }\). Towards the end of June 2008 there was a spillage in the lake which may have caused the mercury level to rise. Because of this the level was then measured at six randomly chosen points of the lake, and the mean of this sample is denoted by \(\bar { x } _ { 2 }\).

1. State hypotheses for a test based on the two samples for whether, on average, the level of mercury had increased. Define any parameters that you use.
2. Find the set of values of \(\bar { x } _ { 2 } - \bar { x } _ { 1 }\) for which there would be evidence at the 5\% significance level that, on average, the level of mercury had increased.
3. Given that the average level had actually increased by 0.3 units, find the probability of making a Type II error in your test, and comment on its value.

6 A mathematics examination is taken by 29 boys and 26 girls. Experience has shown that the probability that any boy forgets to bring a calculator to the examination is 0.3 , and that any girl forgets is 0.2 . Whether or not any student forgets to bring a calculator is independent of all other students. The numbers of boys and girls who forget to bring a calculator are denoted by \(B\) and \(G\) respectively, and \(F = B + G\).

1. Find \(\mathrm { E } ( F )\) and \(\operatorname { Var } ( F )\).
2. Using suitable approximations to the distributions of \(B\) and \(G\), which should be justified, find the smallest number of spare calculators that should be available in order to be at least \(99 \%\) certain that all 55 students will have a calculator.

7 A tutor gives a randomly selected group of 8 students an English Literature test, and after a term's further teaching, she gives the group a similar test. The marks for the two tests are given in the table.

1. Stating a necessary condition, show by carrying out a suitable \(t\)-test, at the \(1 \%\) significance level, that the marks do not give evidence of an improvement.
2. The tutor later found that she had marked the second test too severely, and she decided to add a constant amount \(k\) to each mark. Find the least integer value of \(k\) for which the increased marks would give evidence of improvement at the \(1 \%\) significance level.

8 A soft drinks factory produces lemonade which is sold in packs of 6 bottles. As part of the factory's quality control, random samples of 75 packs are examined at regular intervals. The number of underfilled bottles in a pack of 6 bottles is denoted by the random variable \(X\). The results of one quality control check are shown in the following table. A researcher assumes that \(X \sim \mathrm {~B} ( 3 , p )\).

1. By finding the sample mean, show that an estimate of \(p\) is 0.2 .
2. Show that, at the \(5 \%\) significance level, there is evidence that this binomial distribution does not fit the data.
3. Another researcher suggests that the goodness of fit test should be for \(\mathrm { B } ( 6 , p )\). She finds that the corresponding value of \(\chi ^ { 2 }\) is 2.74 , correct to 3 significant figures. Given that the number of degrees of freedom is the same as in part (ii), state the conclusion of the test at the same significance level.