OCR S3 (Statistics 3) 2014 June

1 The independent random variables \(X\) and \(Y\) have Poisson distributions with parameters 16 and 2 respectively, and \(Z = \frac { 1 } { 2 } X - Y\).

1. Find \(\mathrm { E } ( Z )\) and \(\operatorname { Var } ( Z )\).
2. State whether \(Z\) has a Poisson distribution, giving a reason for your answer.

2 In a study of the inheritance of skin colouration in corn snakes, a researcher found 865 snakes with black and orange bodies, 320 snakes with black bodies, 335 snakes with orange bodies and 112 snakes with bodies of other colours. Theory predicts that snakes of these colours should occur in the ratios \(9 : 3 : 3 : 1\). Test, at the \(5 \%\) significance level, whether these experimental results are compatible with theory.

3 An athlete finds that her times for running 100 m are normally distributed. Before a period of intensive training, her mean time is 11.8 s . After the period of intensive training, five randomly selected times, in seconds, are as follows. $$\begin{array} { l l l l l } 11.70 & 11.65 & 11.80 & 11.75 & 11.60 \end{array}$$ Carry out a suitable test, at the \(5 \%\) significance level, to investigate whether times after the training are less, on average, than times before the training.

4 Cola is sold in bottles and cans. The volume of cola in a bottle is normally distributed with mean 500 ml and standard deviation 10 ml . The volume of cola in a can is normally distributed with mean 330 ml and standard deviation 8 ml . Find the probability that the total volume of cola in 2 randomly selected bottles is greater than 3 times the volume of cola in a randomly selected can.

5 The day before the 1992 General Election, an opinion poll showed that \(37.6 \%\) of a random sample of 1731 voters intended to vote for the Conservative party.

1. Calculate an approximate \(99.9 \%\) confidence interval for the proportion of voters intending to vote Conservative. The actual proportion voting Conservative was above the upper limit of the confidence interval.
2. Give two possible reasons for this occurrence.
3. What sample size would be required to produce a \(99.9 \%\) confidence interval of width 0.05 ?

6 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k \sin x & 0 \leqslant x \leqslant \frac { 1 } { 2 } \pi ,
k \left( 2 - \frac { 2 x } { \pi } \right) & \frac { 1 } { 2 } \pi \leqslant x \leqslant \pi ,
0 & \text { otherwise, } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 4 } { 4 + \pi }\).
2. Find \(\mathrm { E } ( X )\), correct to 3 significant figures, showing all necessary working.

7 A random sample of 100 adults with a chronic disease was chosen. Each adult was randomly assigned to one of three different treatments. After six months of treatment, each adult was then assessed and classified as 'much improved', 'improved', 'slightly improved' or 'no change'. The results are summarised in Table 1. \begin{table}[h] \end{table} A \(\chi ^ { 2 }\) test, at the \(5 \%\) significance level, is to be carried out.

1. State suitable hypotheses. Combining the last two rows of Table 1 gives Table 2. \begin{table}[h]

   	Treatment \(A\)	Treatment \(B\)	Treatment \(C\)
   Much improved	12	16	4
   Improved	13	12	6
   Slightly improved/ No change	12	9	16

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. By considering the expected frequencies for Treatment \(C\) in Table 1, explain why it was necessary to combine rows.
3. Show that the contribution to the \(\chi ^ { 2 }\) value for the cell 'slightly improved/no change, Treatment \(C\) ' is 4.231 , correct to 3 decimal places. You are given that the \(\chi ^ { 2 }\) test statistic is 10.51 , correct to 2 decimal places.
4. Carry out the test.

8 A random sample of 20 plots of land, each of equal area, was used to test whether the addition of phosphorus would increase the yield of corn. 10 plots were treated with phosphorus and 10 plots were untreated. The yields of corn, in litres, on a treated plot and on an untreated plot are denoted by \(X\) and \(Y\) respectively. You are given that $$\sum x = 2112 , \quad \sum y = 2008$$ You are also given that an unbiased estimate for the variance of treated plots is 87.96 and an unbiased estimate for the variance of untreated plots is 31.96 , both correct to 4 significant figures.

1. You may assume that the population variance estimates are sufficiently similar for the assumption of common variance to be made. What other assumption needs to be made for a \(t\)-test to be valid?
2. Carry out a suitable \(t\)-test at the \(1 \%\) significance level, to test whether the use of phosphorus increases the yield of corn.

9 A rectangle of area \(A \mathrm {~m} ^ { 2 }\) has a perimeter of 20 m and each of the two shorter sides are of length \(X \mathrm {~m}\), where \(X\) is uniformly distributed between 0 and 2 .

1. Write down an expression for \(A\) in terms of \(X\), and hence show that \(A = 25 - ( X - 5 ) ^ { 2 }\).
2. Write down the probability density function of \(X\).
3. Show that the cumulative distribution function of \(A\) is $$\mathrm { F } ( a ) = \left\{ \begin{array} { l r } 0 & a < 0 ,
   \frac { 1 } { 2 } ( 5 - \sqrt { 25 - a } ) & 0 \leqslant a \leqslant 16 ,
   1 & a > 16 . \end{array} \right.$$
4. Find the probability density function of \(A\). \section*{END OF QUESTION PAPER} \section*{\(\mathrm { OCR } ^ { \text {勾 } }\)}