OCR MEI S2 (Statistics 2) 2010 January

1 A pilot records the take-off distance for his light aircraft on runways at various altitudes. The data are shown in the table below, where \(a\) metres is the altitude and \(t\) metres is the take-off distance. Also shown are summary statistics for these data. $$n = 7 \quad \Sigma a = 6300 \quad \Sigma t = 5987 \quad \Sigma a ^ { 2 } = 8190000 \quad \Sigma t ^ { 2 } = 5288931 \quad \Sigma a t = 6037800$$

1. Draw a scatter diagram to illustrate these data.
2. State which of the two variables \(a\) and \(t\) is the independent variable and which is the dependent variable. Briefly explain your answer.
3. Calculate the equation of the regression line of \(t\) on \(a\).
4. Use the equation of the regression line to calculate estimates of the take-off distance for altitudes
   (A) 800 metres,
   (B) 2500 metres. Comment on the reliability of each of these estimates.
5. Calculate the value of the residual for the data point where \(a = 1200\) and \(t = 923\), and comment on its sign.

2 On average 2\% of a particular model of laptop computer are faulty. Faults occur independently and randomly.

1. Find the probability that exactly 1 of a batch of 10 laptops is faulty.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. A school buys a batch of 150 of these laptops. Use a Poisson approximating distribution to find the probability that
   (A) there are no faulty laptops in the batch,
   (B) there are more than the expected number of faulty laptops in the batch.
4. A large company buys a batch of 2000 of these laptops for its staff.
   (A) State the exact distribution of the number of faulty laptops in this batch.
   (B) Use a suitable approximating distribution to find the probability that there are at most 50 faulty laptops in this batch.

3 In an English language test for 12-year-old children, the raw scores, \(X\), are Normally distributed with mean 45.3 and standard deviation 11.5.

1. Find
   (A) \(\mathrm { P } ( X < 50 )\),
   (B) \(\mathrm { P } ( 45.3 < X < 50 )\).
2. Find the least raw score which would be obtained by the highest scoring \(10 \%\) of children.
3. The raw score is then scaled so that the scaled score is Normally distributed with mean 100 and standard deviation 15. This scaled score is then rounded to the nearest integer. Find the probability that a randomly selected child gets a rounded score of exactly 111 .
4. In a Mathematics test for 12-year-old children, the raw scores, \(Y\), are Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(\mathrm { P } ( Y < 15 ) = 0.3\) and \(\mathrm { P } ( Y < 22 ) = 0.8\), find the values of \(\mu\) and \(\sigma\).

4 A council provides waste paper recycling services for local businesses. Some businesses use the standard service for recycling paper, others use a special service for dealing with confidential documents, and others use both. Businesses are classified as small or large. A survey of a random sample of 285 businesses gives the following data for size of business and recycling service.

1. Write down null and alternative hypotheses for a test to examine whether there is any association between size of business and recycling service used. The contributions to the test statistic for the usual \(\chi ^ { 2 }\) test are shown in the table below.

   		Recycling Service
   \cline { 3 - 5 } \multicolumn{2}{|c|}{}	Standard	Special	Both
   \multirow{2}{*}{ Size of business }	Small	0.1023	0.2607	0.0186
   	Large	0.0597	0.1520	0.0108

   The sum of these contributions is 0.6041 .
2. Calculate the expected frequency for large businesses using the special service. Verify the corresponding contribution 0.1520 to the test statistic.
3. Carry out the test at the \(5 \%\) level of significance, stating your conclusion clearly. The council is also investigating the weight of rubbish in domestic dustbins. In 2008 the average weight of rubbish in bins was 32.8 kg . The council has now started a recycling initiative and wishes to determine whether there has been a reduction in the weight of rubbish in bins. A random sample of 50 domestic dustbins is selected and it is found that the mean weight of rubbish per bin is now 30.9 kg , and the standard deviation is 3.4 kg .
4. Carry out a test at the \(5 \%\) level to investigate whether the mean weight of rubbish has been reduced in comparison with 2008 . State carefully your null and alternative hypotheses. {www.ocr.org.uk}) after the live examination series.
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