AQA S3 (Statistics 3) 2010 June

2 Rodney and Derrick, two independent fruit and vegetable market stallholders, sell punnets of locally-grown raspberries from their stalls during June and July. The following information, based on independent random samples, was collected as part of an investigation by Trading Standards Officers.

1. Construct a \(99 \%\) confidence interval for the difference between the mean weight of raspberries in a punnet sold by Rodney and the mean weight of raspberries in a punnet sold by Derrick.
2. What can be concluded from your confidence interval?
3. In addition to weight, state one other factor that may influence whether customers buy raspberries from Rodney or from Derrick.
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3
The weekly number of hits, \(S\), on Sam's website may be modelled by a Poisson distribution with parameter \(\lambda _ { S }\). The weekly number of hits, \(T\), on Tina's website may be modelled by a Poisson distribution with parameter \(\lambda _ { T }\).
During a period of 40 weeks, the number of hits on Sam's website was 940.
During a period of 60 weeks, the number of hits on Tina's website was 1560.
Assuming that \(S\) and \(T\) are independent random variables, investigate, at the \(2 \%\) level of significance, Tina's claim that the mean weekly number of hits on her website is greater than that on Sam's website.
(7 marks)
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4 It is proposed to introduce, for all males at age 60, screening tests, A and B, for a certain disease. Test B is administered only when the result of Test A is inconclusive. It is known that 10\% of 60-year-old men suffer from the disease. For those 60 -year-old men suffering from the disease:

• Test A is known to give a positive result, indicating a presence of the disease, in \(90 \%\) of cases, a negative result in \(2 \%\) of cases and a requirement for the administration of Test B in \(8 \%\) of cases;
• Test B is known to give a positive result in \(98 \%\) of cases and a negative result in 2\% of cases.

For those 60 -year-old men not suffering from the disease:

• Test A is known to give a positive result in \(1 \%\) of cases, a negative result in \(80 \%\) of cases and a requirement for the administration of Test B in 19\% of cases;
• Test B is known to give a positive result in \(1 \%\) of cases and a negative result in \(99 \%\) of cases.
  1. Draw a tree diagram to represent the above information.
  2. 1. Hence, or otherwise, determine the probability that:
        (A) a 60-year-old man, suffering from the disease, tests negative;
        (B) a 60-year-old man, not suffering from the disease, tests positive.
     2. A random sample of ten thousand 60-year-old men is given the screening tests. Calculate, to the nearest 10, the number who you would expect to be given an incorrect diagnosis.
  3. Determine the probability that:
     1. a 60-year-old man suffers from the disease given that the tests provide a positive result;
     2. a 60-year-old man does not suffer from the disease given that the tests provide a negative result.

5 In the manufacture of desk drawer fronts, a machine cuts sheets of veneered chipboard into rectangular pieces of width \(W\) millimetres and height \(H\) millimetres. The 4 edges of each of these pieces are then covered with matching veneered tape. The distributions of \(W\) and \(H\) are such that $$\mathrm { E } ( W ) = 350 \quad \operatorname { Var } ( W ) = 5 \quad \mathrm { E } ( H ) = 210 \quad \operatorname { Var } ( H ) = 4 \quad \rho _ { W H } = 0.75$$

1. Calculate the mean and the variance of the length of tape, \(T = 2 W + 2 H\), needed for the edges of a drawer front.
2. A desk has 4 such drawers whose sizes may be assumed to be independent. Given that \(T\) may be assumed to be normally distributed, determine the probability that the total length of tape needed for the edges of the desk's 4 drawer fronts does not exceed 4.5 metres.
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6

1. A district council claimed that more than 80 per cent of the complaints that it received about the delivery of its services were answered to the satisfaction of complainants before reaching formal status. An analysis of a random sample of 175 complaints revealed that 28 reached formal status.
   1. Construct an approximate \(95 \%\) confidence interval for the proportion of complaints that reach formal status.
   2. Hence comment on the council's claim.
2. The district council also claimed that less than 40 per cent of all formal complaints were due to a failing in the delivery of its services. An analysis of the 50 formal complaints received during 2007/08 showed that 16 were due to a failing in the delivery of its services.
   1. Using an exact test, investigate the council's claim at the \(10 \%\) level of significance. The 50 formal complaints received during 2007/08 may be assumed to be a random sample.
   2. Determine the critical value for your test in part (b)(i).
   3. In fact, only 25 per cent of all formal complaints were due to a failing in the delivery of the council's services. Determine the probability of a Type II error for a test of the council's claim at the \(10 \%\) level of significance and based on the analysis of a random sample of 50 formal complaints.
      (4 marks)
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7 The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).

1. 1. Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
   2. Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }\), find, in terms of \(\lambda\), an expression for \(\operatorname { Var } ( X )\).
2. The mode, \(m\), of \(X\) is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
   1. Show that \(\lambda - 1 \leqslant m \leqslant \lambda\).
   2. Given that \(\lambda = 4.9\), determine \(\mathrm { P } ( X = m )\).
3. The random variable \(Y\) has a Poisson distribution with mode \(d\) and standard deviation 15.5. Use a distributional approximation to estimate \(\mathrm { P } ( Y \geqslant d )\).
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