OCR MEI S2 (Statistics 2) 2011 January

1 The scatter diagram below shows the birth rates \(x\), and death rates \(y\), measured in standard units, in a random sample of 14 countries in a particular year. Summary statistics for the data are as follows. $$\Sigma x = 139.8 \quad \Sigma y = 140.4 \quad \Sigma x ^ { 2 } = 1411.66 \quad \Sigma y ^ { 2 } = 1417.88 \quad \Sigma x y = 1398.56 \quad n = 14$$ \includegraphics[max width=\textwidth, alt={}, center]{cd1a8f39-dd3c-44c9-90b0-6a919361d593-2_643_1047_488_550}

1. Calculate the sample product moment correlation coefficient.
2. Carry out a hypothesis test at the \(5 \%\) significance level to determine whether there is any correlation between birth rates and death rates.
3. State the distributional assumption which is necessary for this test to be valid. Explain briefly in the light of the scatter diagram why it appears that the assumption may be valid.
4. The values of \(x\) and \(y\) for another country in that year are 14.4 and 7.8 respectively. If these values are included, the value of the sample product moment correlation coefficient is - 0.5694 . Explain why this one observation causes such a large change to the value of the sample product moment correlation coefficient. Discuss whether this brings the validity of the test into question.

2 A student is investigating the numbers of sultanas in a particular brand of biscuit. The data in the table show the numbers of sultanas in a random sample of 50 of these biscuits.

1. Show that the sample mean is 1.84 and calculate the sample variance.
2. Explain why these results support a suggestion that a Poisson distribution may be a suitable model for the distribution of the numbers of sultanas in this brand of biscuit. For the remainder of the question you should assume that a Poisson distribution with mean 1.84 is a suitable model for the distribution of the numbers of sultanas in these biscuits.
3. Find the probability of
   (A) no sultanas in a biscuit,
   (B) at least two sultanas in a biscuit.
4. Show that the probability that there are at least 10 sultanas in total in a packet containing 5 biscuits is 0.4389 .
5. Six packets each containing 5 biscuits are selected at random. Find the probability that exactly 2 of the six packets contain at least 10 sultanas.
6. Sixty packets each containing 5 biscuits are selected at random. Use a suitable approximating distribution to find the probability that more than half of the sixty packets contain at least 10 sultanas.

3 The random variable \(X\) represents the reaction times, in milliseconds, of men in a driving simulator. \(X\) is Normally distributed with mean 355 and standard deviation 52.

1. Find
   (A) \(\mathrm { P } ( X < 325 )\),
   (B) \(\mathrm { P } ( 300 < X < 400 )\).
2. Find the value of \(k\) for which \(\mathrm { P } ( X < k ) = 0.2\). It is thought that women may have a different mean reaction time from men. In order to test this, a random sample of 25 women is selected. The mean reaction time of these women in the driving simulator is 344 milliseconds. You may assume that women's reaction times are also Normally distributed with standard deviation 52 milliseconds. A hypothesis test is carried out to investigate whether women have a different mean reaction time from men.
3. Carry out the test at the \(5 \%\) significance level.

4 A researcher is investigating the sizes of pebbles at various locations in a river. Three sites in the river are chosen and each pebble sampled at each site is classified as large, medium or small. The results are as follows.

1. Carry out a test at the \(5 \%\) significance level to examine whether there is any association between pebble size and site. Your working should include a table of the contributions of each cell to the test statistic.
2. By referring to each site, comment briefly on how the size of the pebbles compares with what would be expected if there were no association. You should support your answers by referring to your table of contributions.