OCR MEI S2 (Statistics 2) 2009 June

1 An investment analyst thinks that there may be correlation between the cost of oil, \(x\) dollars per barrel, and the price of a particular share, \(y\) pence. The analyst selects 50 days at random and records the values of \(x\) and \(y\). Summary statistics for these data are shown below, together with a scatter diagram. $$\Sigma x = 2331.3 \quad \Sigma y = 6724.3 \quad \Sigma x ^ { 2 } = 111984 \quad \Sigma y ^ { 2 } = 921361 \quad \Sigma x y = 316345 \quad n = 50$$ \includegraphics[max width=\textwidth, alt={}, center]{ae79cdd9-a57c-490e-a9f3-f47c7c8a1aa6-2_857_905_516_621}

1. Calculate the sample product moment correlation coefficient.
2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate the analyst's belief. State your hypotheses clearly, defining any symbols which you use.
3. An assumption that there is a bivariate Normal distribution is required for this test to be valid. State whether it is the sample or the population which is required to have such a distribution. State, with a reason, whether in this case the assumption appears to be justified.
4. Explain why a 2-tail test is appropriate even though it is clear from the scatter diagram that the sample has a positive correlation coefficient.

2 Jess is watching a shower of meteors (shooting stars). During the shower, she sees meteors at an average rate of 1.3 per minute.

1. State conditions required for a Poisson distribution to be a suitable model for the number of meteors which Jess sees during a randomly selected minute. You may assume that these conditions are satisfied.
2. Find the probability that, during one minute, Jess sees
   (A) exactly one meteor,
   (B) at least 4 meteors.
3. Find the probability that, in a period of 10 minutes, Jess sees exactly 10 meteors.
4. Use a suitable approximating distribution to find the probability that Jess sees a total of at least 100 meteors during a period of one hour.
5. Jess watches the shower for \(t\) minutes. She wishes to be at least \(99 \%\) certain that she will see one or more meteors. Find the smallest possible integer value of \(t\).

3 Intensity of light is measured in lumens. The random variable \(X\) represents the intensity of the light from a standard 100 watt light bulb. \(X\) is Normally distributed with mean 1720 and standard deviation 90. You may assume that the intensities for different bulbs are independent.

1. Show that \(\mathrm { P } ( X < 1700 ) = 0.4121\).
2. These bulbs are sold in packs of 4 . Find the probability that the intensities of exactly 2 of the 4 bulbs in a randomly chosen pack are below 1700 lumens.
3. Use a suitable approximating distribution to find the probability that the intensities of at least 20 out of 40 randomly selected bulbs are below 1700 lumens. A manufacturer claims that the average intensity of its 25 watt low energy light bulbs is 1720 lumens. A consumer organisation suspects that the true figure may be lower than this. The intensities of a random sample of 20 of these bulbs are measured. A hypothesis test is then carried out to check the claim.
4. Write down a suitable null hypothesis and explain briefly why the alternative hypothesis should be \(\mathrm { H } _ { 1 } : \mu < 1720\). State the meaning of \(\mu\).
5. Given that the standard deviation of the intensity of such bulbs is 90 lumens and that the mean intensity of the sample of 20 bulbs is 1703 lumens, carry out the test at the \(5 \%\) significance level.

4 In a traffic survey a random sample of 400 cars passing a particular location during the rush hour is selected. The type of car and the sex of the driver are classified as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between type of car and sex of driver. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.
2. For each type of car, comment briefly on how the number of drivers of each sex compares with what would be expected if there were no association. 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, is given to all schools that receive assessment material 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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