OCR MEI S1 (Statistics 1) 2011 June

1 In the Paris-Roubaix cycling race, there are a number of sections of cobbled road. The lengths of these sections, measured in metres, are illustrated in the histogram.
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1. Find the number of sections which are between 1000 and 2000 metres in length.
2. Name the type of skewness suggested by the histogram.
3. State the minimum and maximum possible values of the midrange.

2 I have 5 books, each by a different author. The authors are Austen, Brontë, Clarke, Dickens and Eliot.

1. If I arrange the books in a random order on my bookshelf, find the probability that the authors are in alphabetical order with Austen on the left.
2. If I choose two of the books at random, find the probability that I choose the books written by Austen and Brontë.
   \(325 \%\) of the plants of a particular species have red flowers. A random sample of 6 plants is selected.
3. Find the probability that there are no plants with red flowers in the sample.
4. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers.

4 Two fair six-sided dice are thrown. The random variable \(X\) denotes the difference between the scores on the two dice. The table shows the probability distribution of \(X\).

1. Draw a vertical line chart to illustrate the probability distribution.
2. Use a probability argument to show that
   (A) \(\mathrm { P } ( X = 1 ) = \frac { 5 } { 18 }\),
   (B) \(\mathrm { P } ( X = 0 ) = \frac { 1 } { 6 }\).
3. Find the mean value of \(X\).

6 The numbers of eggs laid by a sample of 70 female herring gulls are shown in the table.

1. Find the mean and standard deviation of the number of eggs laid per gull.
2. The sample did not include female herring gulls that laid no eggs. How would the mean and standard deviation change if these gulls were included?

7 Any patient who fails to turn up for an outpatient appointment at a hospital is described as a 'no-show'. At a particular hospital, on average \(15 \%\) of patients are no-shows. A random sample of 20 patients who have outpatient appointments is selected.

1. Find the probability that
   (A) there is exactly 1 no-show in the sample,
   (B) there are at least 2 no-shows in the sample. The hospital management introduces a policy of telephoning patients before appointments. It is hoped that this will reduce the proportion of no-shows. In order to check this, a random sample of \(n\) patients is selected. The number of no-shows in the sample is recorded and a hypothesis test is carried out at the 5\% level.
2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
3. In the case that \(n = 20\) and the number of no-shows in the sample is 1 , carry out the test.
4. In another case, where \(n\) is large, the number of no-shows in the sample is 6 and the critical value for the test is 8 . Complete the test.
5. In the case that \(n \leqslant 18\), explain why there is no point in carrying out the test at the \(5 \%\) level.

8 The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

1. Draw a cumulative frequency diagram to illustrate these data.
2. Use the diagram to estimate the median and interquartile range of the data.
3. Show that there are no outliers in the sample.
4. Three of these 50 sacks are selected at random. Find the probability that
   (A) in all three, the heating quality \(x\) is more than 9.5,
   \(( B )\) in at least two, the heating quality \(x\) is more than 9.5. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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