OCR MEI S1 (Statistics 1) 2007 June

1 A girl is choosing tracks from an album to play at her birthday party. The album has 8 tracks and she selects 4 of them.

1. In how many ways can she select the 4 tracks?
2. In how many different orders can she arrange the 4 tracks once she has chosen them?

2 The histogram shows the amount of money, in pounds, spent by the customers at a supermarket on a particular day.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-2_977_1132_808_340}
□ represents 20 customers

1. Express the data in the form of a grouped frequency table.
2. Use your table to estimate the total amount of money spent by customers on that day.

3 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$

1. Calculate the mean and standard deviation of the marks.
2. The highest mark scored by any of the 56 students in the examination was 93 . Show that this result may be considered to be an outlier.
3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.

4 A local council has introduced a recycling scheme for aluminium, paper and kitchen waste. 50 residents are asked which of these materials they recycle. The numbers of people who recycle each type of material are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-3_803_803_406_671} One of the residents is selected at random.

1. Find the probability that this resident recycles
   (A) at least one of the materials,
   (B) exactly one of the materials.
2. Given that the resident recycles aluminium, find the probability that this resident does not recycle paper. Two residents are selected at random.
3. Find the probability that exactly one of them recycles kitchen waste.

5 A GCSE geography student is investigating a claim that global warming is causing summers in Britain to have more rainfall. He collects rainfall data from a local weather station for 2001 and 2006. The vertical line chart shows the number of days per week on which some rainfall was recorded during the 22 weeks of summer 2001.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-4_720_1557_443_296} Number of days per week with rain recorded in summer 2001

1. Show that the median of the data is 4 , and find the interquartile range.
2. For summer 2006 the median is 3 and the interquartile range is also 3. The student concludes that the data demonstrate that global warming is causing summer rainfall to decrease rather than increase. Is this a valid conclusion from the data? Give two brief reasons to justify your answer.

6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.

1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.

   \(r\)	0	1	2	3	4
   \(\mathrm { P } ( X = r )\)	0	\(\frac { 4 } { 35 }\)	\(\frac { 18 } { 35 }\)	\(\frac { 12 } { 35 }\)	\(\frac { 1 } { 35 }\)

2. Find the expectation and variance of \(X\).

7 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, \(3 \%\) is classified as positive, \(6 \%\) as doubtful and the rest negative. In fact, of the people who test positive, only \(95 \%\) have the disease. Of the people who test doubtful, \(10 \%\) have the disease. Of the people who test negative, \(1 \%\) actually have the disease. People who do not have the disease are described as 'clear'.

1. Copy and complete the tree diagram to show this information.
   \includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-5_830_1157_845_536}
2. Find the probability that a randomly selected person tests negative and is clear.
3. Find the probability that a randomly selected person has the disease.
4. Find the probability that a randomly selected person tests negative given that the person has the disease.
5. Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in \(98 \%\) of cases. If a person is clear, the examination will always correctly identify this.
6. A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.

8 A multinational accountancy firm receives a large number of job applications from graduates each year. On average \(20 \%\) of applicants are successful. A researcher in the human resources department of the firm selects a random sample of 17 graduate applicants.

1. Find the probability that at least 4 of the 17 applicants are successful.
2. Find the expected number of successful applicants in the sample.
3. Find the most likely number of successful applicants in the sample, justifying your answer. It is suggested that mathematics graduates are more likely to be successful than those from other fields. In order to test this suggestion, the researcher decides to select a new random sample of 17 mathematics graduate applicants. The researcher then carries out a hypothesis test at the \(5 \%\) significance level.
4. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Give a reason for your choice of the alternative hypothesis.
5. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
6. Explain why the critical region found in part (v) would be unaltered if a \(10 \%\) significance level were used.