OCR MEI S1 (Statistics 1) 2011 June

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. \includegraphics{figure_1}

1. Find the number of sections which are between 1000 and 2000 metres in length. [2]
2. Name the type of skewness suggested by the histogram. [1]
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]
2. If I choose two of the books at random, find the probability that I choose the books written by Austen and Brontë. [3]

25% of the plants of a particular species have red flowers. A random sample of 6 plants is selected.

1. Find the probability that there are no plants with red flowers in the sample. [2]
2. If 50 random samples of 6 plants are selected, find the expected number of samples in which there are no plants with red flowers. [2]

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 \(X\).

\(r\)	0	1	2	3	4	5
P(X = r)	\(\frac{1}{6}\)	\(\frac{5}{18}\)	\(\frac{2}{9}\)	\(\frac{1}{6}\)	\(\frac{1}{9}\)	\(\frac{1}{18}\)

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

In a recent survey, a large number of working people were asked whether they worked full-time or part-time, with part-time being defined as less than 25 hours per week. One of the respondents is selected at random. • \(W\) is the event that this person works part-time. • \(F\) is the event that this person is female. You are given that P(\(W\)) = 0.14, P(\(F\)) = 0.41 and P(\(W \cap F\)) = 0.11.

1. Draw a Venn diagram showing the events \(W\) and \(F\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
2. Determine whether the events \(W\) and \(F\) are independent. [2]
3. Find P(\(W\) | \(F\)) and explain what this probability represents. [3]

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

Number of eggs	1	2	3	4
Frequency	10	40	15	5

1. Find the mean and standard deviation of the number of eggs laid per gull. [5]
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? [2]

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
   1. there is exactly 1 no-show in the sample, [3]
   2. there are at least 2 no-shows in the sample. [2]

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.

1. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis. [4]
2. In the case that \(n = 20\) and the number of no-shows in the sample is 1, carry out the test. [4]
3. 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. [3]
4. In the case that \(n \leqslant 18\), explain why there is no point in carrying out the test at the 5% level. [2]

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

Heating quality (\(x\))	9.1 \(\leqslant x <\) 9.3	9.3 \(< x \leqslant\) 9.5	9.5 \(< x \leqslant\) 9.7	9.7 \(< x \leqslant\) 9.9	9.9 \(< x \leqslant\) 10.1
Frequency	5	7	15	16	7

1. Draw a cumulative frequency diagram to illustrate these data. [5]
2. Use the diagram to estimate the median and interquartile range of the data. [3]
3. Show that there are no outliers in the sample. [3]
4. Three of these 50 sacks are selected at random. Find the probability that
   1. in all three, the heating quality \(x\) is more than 9.5, [3]
   2. in at least two, the heating quality \(x\) is more than 9.5. [4]