OCR S3 (Statistics 3) 2006 June

1 The numbers of \(\alpha\)-particles emitted per minute from two types of source, \(A\) and \(B\), have the distributions \(\operatorname { Po } ( 1.5 )\) and \(\operatorname { Po } ( 2 )\) respectively. The total number of \(\alpha\)-particles emitted over a period of 2 minutes from three sources of type \(A\) and two sources of type \(B\), all of which are independent, is denoted by \(X\). Calculate \(\mathrm { P } ( X = 27 )\).

2 The manager of a factory with a large number of employees investigated when accidents to employees occurred during 8-hour shifts. An analysis was made of 600 randomly chosen accidents that occurred over a year. The following table shows the numbers of accidents occurring in the four consecutive 2-hour periods of the 8-hour shifts. Test, at the \(5 \%\) significance level, whether the proportions of all accidents that occur in the four time periods differ.

3 Ten randomly chosen athletes were coached for a 200 m event. For each athlete, the times taken to run 200 m before and after coaching were measured. The sample mean times before and after coaching were 23.43 seconds and 22.84 seconds respectively. For each athlete the difference, \(d\) seconds, in the times before and after coaching was calculated and an unbiased estimate of the population variance of \(d\) was found to be 0.548 . Stating any required assumption, test at the \(5 \%\) significance level whether the population mean time for the 200 m run decreased after coaching.

4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 4 } { 3 x ^ { 3 } } & 1 \leqslant x < 2
\frac { 1 } { 12 } x & 2 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

1. Find the upper quartile of \(X\).
2. Find the value of \(a\) for which \(\mathrm { E } \left( X ^ { 2 } \right) = a \mathrm { E } ( X )\).

5 Gloria is a market trader who sells jeans. She trades on Mondays, Wednesdays and Fridays. Wishing to investigate whether the volume of trade depends on the day of the week, Gloria analysed a random sample of 150 days' sales and classified them by day and volume (low, medium and high). The results are given in the table below. Gloria asked a statistician to perform a suitable test of independence and, as part of this test, expected frequencies were calculated. These are shown in the table below.

1. Show how the value 23.04 for medium volume on Wednesday has been obtained.
2. State, giving a reason, if it is necessary to combine any rows or columns in order to carry out the test. The value of the test statistic is found to be 21.15, correct to 2 decimal places.
3. Stating suitable hypotheses for the test, give its conclusion using a \(1 \%\) significance level. Gloria wishes to hold a sale and asks the statistician to advise her on which day to hold it in order to sell as much as possible.
4. State the day that the statistician should advise and give a reason for the choice.

6 An anthropologist was studying the inhabitants of two islands, Raloa and Tangi. Part of the study involved the incidence of blood group type A. The blood of 80 randomly chosen inhabitants of Raloa and 85 randomly chosen inhabitants of Tangi was tested. The number of inhabitants with type A blood was 28 for the Raloa sample and 46 for the Tangi sample. The anthropologist calculated \(90 \%\) confidence intervals for the population proportions of inhabitants with type A blood. They were \(( 0.262,0.438 )\) for Raloa and \(( 0.452,0.630 )\) for Tangi, where each figure is correct to 3 decimal places. It is known that \(43 \%\) of the world's population have type A blood.

1. State, giving your reasons, whether there is evidence for the following assertions about the proportions of people with type A blood.
   (a) The proportion in Raloa is different from the world proportion.
   (b) The proportion in Tangi is different from the world proportion.
2. Carry out a suitable test, at the \(2 \%\) significance level, of whether the proportions of people with type A blood differ on the two islands.

7 A queue of cars has built up at a set of traffic lights which are at red. When the lights turn green, the time for the first car to start to move has a normal distribution with mean 2.2 s and standard deviation 0.75 s . This time is the reaction time for the first car. For each subsequent car the reaction time is the time taken for it to start to move after the car in front starts to move. These reaction times have identical normal distributions with mean 1.8 s and standard deviation 0.70 s . It may be assumed that all reaction times are independent.

1. Calculate the probability that the reaction time for the second car in the queue is less than half of the reaction time for the first car.
2. Calculate the probability that the fifth car in the queue starts to move less than 10 seconds after the lights turn green.
3. State where, in part (i), independence is required.

8 Two machines, \(A\) and \(B\), produce metal rods. Machine \(B\) is new and it is required that its accuracy should be checked against that of machine \(A\). The observed variable is the length of a rod. Random samples of rods, 40 from machine \(A\) and 50 from machine \(B\), are taken and their lengths, \(x _ { A } \mathrm {~cm}\) and \(x _ { B } \mathrm {~cm}\), are measured. The results are summarised by $$\Sigma x _ { A } = 136.48 , \quad \Sigma x _ { B } = 176.35 , \quad \Sigma x _ { B } ^ { 2 } = 630.1940 .$$ The variance of the length of the rods produced by machine \(A\) is known to be \(0.0490 \mathrm {~cm} ^ { 2 }\). The mean lengths of the rods produced by the machines are denoted by \(\mu _ { A } \mathrm {~cm}\) and \(\mu _ { B } \mathrm {~cm}\) respectively.

1. Test, at the \(5 \%\) significance level, the hypothesis \(\mu _ { B } > \mu _ { A }\).
2. Find the set of values of \(a\) for which the null hypothesis \(\mu _ { B } - \mu _ { A } = 0.025\) would not be rejected in favour of the alternative hypothesis \(\mu _ { B } - \mu _ { A } > 0.025\) at the \(a \%\) significance level.
3. For the test in part (i) to be valid,
   (a) state whether it is necessary to assume that the two population variances are equal,
   (b) state, giving a reason, whether it is necessary to assume that the lengths of rods are normally distributed.