AQA S3 (Statistics 3) 2012 June

1 A wildlife expert measured the neck lengths, \(x\) metres, and the tail lengths, \(y\) metres, of a sample of 12 mature male giraffes as part of a study into their physical characteristics. The results are shown in the table.

2 As part of a comparison of two varieties of cucumber, Fanfare and Marketmore, random samples of harvested cucumbers of each variety were selected and their lengths measured, in centimetres. The results are summarised in the table.

1. Test, at the \(1 \%\) level of significance, the hypothesis that there is no difference between the mean length of harvested Fanfare cucumbers and that of harvested Marketmore cucumbers.
2. In addition to length, name one other characteristic of cucumbers that could be used for comparative purposes.

3 A hotel has three types of room: double, twin and suite. The percentage of rooms in the hotel of each type is 40,45 and 15 respectively. Each room in the hotel may be occupied by \(0,1,2\), or 3 or more people. The proportional occupancy of each type of room is shown in the table.

4 The manager of a medical centre suspects that patients using repeat prescriptions were requesting, on average, more items during 2011 than during 2010. The mean number of items on a repeat prescription during 2010 was 2.6.
An analysis of a random sample of 250 repeat prescriptions during 2011 showed a total of 688 items requested. The number of items requested on a repeat prescription may be modelled by a Poisson distribution. Use a distributional approximation to investigate, at the \(5 \%\) level of significance, the manager's suspicion.

5 A random sample of 125 people was selected from a council's electoral roll. Of these, 68 were in favour of a proposed local building plan.

1. Construct an approximate 98\% confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal.
2. Calculate, to the nearest 5, an estimate of the minimum sample size necessary in order that an approximate \(98 \%\) confidence interval for the percentage of people on the council's electoral roll who were in favour of the proposal has a width of at most 10 per cent.

6 Alyssa lives in the country but works in a city centre.
Her journey to work each morning involves a car journey, a walk and wait, a train journey, and a walk. Her car journey time, \(U\) minutes, from home to the village car park has a mean of 13 and a standard deviation of 3 . Her time, \(V\) minutes, to walk from the village car park to the village railway station and wait for a train to depart has a mean of 15 and a standard deviation of 6 . Her train journey time, \(W\) minutes, from the village railway station to the city centre railway station has a mean of 24 and a standard deviation of 4 . Her time, \(X\) minutes, to walk from the city centre railway station to her office has a mean of 9 and a standard deviation of 2 . The values of the product moment correlation coefficient for the above 4 variables are $$\rho _ { U V } = - 0.6 \quad \text { and } \quad \rho _ { U W } = \rho _ { U X } = \rho _ { V W } = \rho _ { V X } = \rho _ { W X } = 0$$

1. Determine values for the mean and the variance of:
   1. \(M = U + V\);
   2. \(D = W - 2 U\);
   3. \(T = M + W + X\), given that \(\rho _ { M W } = \rho _ { M X } = 0\).
2. Assuming that the variables \(M , D\) and \(T\) are normally distributed, determine the probability that, on a particular morning:
   1. Alyssa's journey time from leaving home to leaving the village railway station is exactly 30 minutes;
   2. Alyssa's train journey time is more than twice her car journey time;
   3. Alyssa's total journey time is between 50 minutes and 70 minutes.

7

1. The random variable \(X\) has a binomial distribution with parameters \(n\) and \(p\).
   1. Prove, from first principles, that \(\mathrm { E } ( X ) = n p\).
   2. Hence, given that \(\mathrm { E } ( X ( X - 1 ) ) = n ( n - 1 ) p ^ { 2 }\), find, in terms of \(n\) and \(p\), an expression for \(\operatorname { Var } ( X )\).
2. The mode, \(m\), of \(X\) is such that $$\mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m - 1 ) \quad \text { and } \quad \mathrm { P } ( X = m ) \geqslant \mathrm { P } ( X = m + 1 )$$
   1. Use the first inequality to show that $$m \leqslant ( n + 1 ) p$$
   2. Given that the second inequality results in $$m \geqslant ( n + 1 ) p - 1$$ deduce that the distribution \(\mathrm { B } ( 10,0.65 )\) has one mode, and find the two values for the mode of the distribution \(B ( 35,0.5 )\).
3. The random variable \(Y\) has a binomial distribution with parameters 4000 and 0.00095 . Use a distributional approximation to estimate \(\mathrm { P } ( Y \leqslant k )\), where \(k\) denotes the mode of \(Y\).
   (3 marks)