OCR MEI S2 (Statistics 2) 2008 January

1 A biology student is carrying out an experiment to study the effect of a hormone on the growth of plant shoots. The student applies the hormone at various concentrations to a random sample of twelve shoots and measures the growth of each shoot. The data are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables \(x\) and \(y\), measured in suitable units, represent concentration and growth respectively.
\includegraphics[max width=\textwidth, alt={}, center]{20fc4222-95c6-4b59-8e89-913dd988eb44-2_693_897_534_625} $$n = 12 , \Sigma x = 30 , \Sigma y = 967.6 , \Sigma x ^ { 2 } = 90 , \Sigma y ^ { 2 } = 78926 , \Sigma x y = 2530.3 .$$

1. State which of the two variables \(x\) and \(y\) is the independent variable and which is the dependent variable. Briefly explain your answers.
2. Calculate the equation of the regression line of \(y\) on \(x\).
3. Use the equation of the regression line to calculate estimates of shoot growth for concentrations of
   (A) 1.2,
   (B) 4.3. Comment on the reliability of each of these estimates.
4. Calculate the value of the residual for the data point where \(x = 3\) and \(y = 80\).
5. In further experiments, the student finds that using concentration \(x = 6\) results in shoot growths of around \(y = 20\). In the light of all the available information, what can be said about the relationship between \(x\) and \(y\) ?

2 A large hotel has 90 bedrooms. Sometimes a guest makes a booking for a room, but then does not arrive. This is called a 'no-show'. On average \(10 \%\) of bookings are no-shows. The hotel manager accepts up to 94 bookings before saying that the hotel is full. If at least 4 of these bookings are no-shows then there will be enough rooms for all of the guests. 94 bookings have been made for each night in August. You should assume that all bookings are independent.

1. State the distribution of the number of no-shows on one night in August.
2. State the conditions under which the use of a Poisson distribution is appropriate as an approximation to a binomial distribution.
3. Use a Poisson approximating distribution to find the probability that, on one night in August,
   (A) there are exactly 4 no-shows,
   (B) there are enough rooms for all of the guests who do arrive.
4. Find the probability that, on all of the 31 nights in August, there are enough rooms for all of the guests who arrive.
5. (A) In August there are \(31 \times 94 = 2914\) bookings altogether. State the exact distribution of the total number of no-shows during August.
   (B) Use a suitable approximating distribution to find the probability that there are at most 300 no-shows altogether during August.

3 In a large population, the diastolic blood pressure (DBP) of 5-year-old children is Normally distributed with mean 56 and standard deviation 6.5.

1. Find the probability that the DBP of a randomly selected 5-year-old child is between 52.5 and 57.5. The DBP of young adults is Normally distributed with mean 68 and standard deviation 10.
2. A 5-year-old child and a young adult are selected at random. Find the probability that the DBP of one of them is over 62 and the other is under 62.
3. Sketch both distributions on a single diagram.
4. For another age group, the DBP is Normally distributed with mean 82. The DBP of \(12 \%\) of people in this age group is below 62. Find the standard deviation for this age group.

4

1. A researcher believes that there may be some association between a student's sex and choice of certain subjects at A-level. A random sample of 250 A -level students is selected. The table below shows, for each sex, how many study either or both of the two subjects, Mathematics and English.

   	Mathematics only	English only	Both	Neither	Row totals
   Male	38	19	6	32	95
   Female	42	55	9	49	155
   Column totals	80	74	15	81	250

   Carry out a test at the \(5 \%\) significance level to examine whether there is any association between a student's sex and choice of subjects. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [12]
2. Over a long period it has been determined that the mean score of students in a particular English module is 67.4 and the standard deviation is 8.9. A new teaching method is introduced with the aim of improving the results. A random sample of 12 students taught by the new method is selected. Their mean score is found to be 68.3. Carry out a test at the \(10 \%\) level to investigate whether the new method appears to have been successful. State carefully your null and alternative hypotheses. You should assume that the scores are Normally distributed and that the standard deviation is unchanged.