Edexcel S2 (Statistics 2)

1. (a) Briefly describe the difference between a census and a sample survey.
   (b) Illustrate the difference by considering the case of a village council which has to decide whether or not to build a new village hall.

Given that the council decides to use a sample survey,
(c) suggest suitable sampling units.

2. The number of copies of The Statistician that a newsagent sells each week is modelled by a Poisson distribution. On average, he sells 1.5 copies per week.

1. Find the probability that he sells no copies in a particular week.
2. If he stocks 5 copies each week, find the probability he will not have enough copies to meet that week's demand.
3. Find the minimum number of copies that he should stock in order to have at least a \(95 \%\) probability of being able to satisfy the week's demand.

3. A die is rolled 60 times, and results in 16 sixes.

1. Use a suitable approximation to test, at the \(5 \%\) significance level, whether the probability of scoring a six is \(\frac { 1 } { 6 }\) or not. State your hypotheses clearly.
2. Describe how you would change the test if you wished to investigate whether the probability of scoring a six is greater than \(\frac { 1 } { 6 }\). Carry out this modified test.

4. A continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { l } 0
\frac { 1 } { 84 } \left( x ^ { 2 } - 16 \right)
1 \end{array} \right.$$ $$\begin{aligned} & x < 4 ,
& 4 \leq x \leq 10 ,
& x > 10 . \end{aligned}$$

1. Find the median value of \(X\).
2. Find the interquartile range for \(X\).
3. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
4. Sketch the graph of \(\mathrm { f } ( x )\) and hence write down the mode of \(X\), explaining how you obtain your answer from the graph. \section*{STATISTICS 2 (A) TEST PAPER 1 Page 2}

1. Lupin seeds are sold in packets of 15 . On average, 9 seeds in a packet are green and 6 are red. Find, to 2 decimal places, the probability that in any particular packet there are
   1. less than 2 red seeds,
   2. more red than green seeds.
   The seeds from 10 packets are then combined together.
2. Use a suitable approximation to find the probability that the total number of green seeds is more than 100 .

6. Patients suffering from 'flu are treated with a drug. The number of days, \(t\), that it then takes for them to recover is modelled by the continuous random variable \(T\) with the probability density function $$\begin{array} { l l } \mathrm { f } ( t ) = \frac { 3 t ^ { 2 } ( 4 - t ) } { 64 } & 0 \leq t \leq 4
\mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$

1. Find the mean and standard deviation of \(T\).
2. Find the probability that a patient takes more than 3 days to recover.
3. Two patients are selected at random. Find the probability that they both recover within three days.
4. Comment on the suitability of the model.

7. In a certain field, daisies are randomly distributed, at an average density of 0.8 daisies per \(\mathrm { cm } ^ { 2 }\). One particular patch, of area \(1 \mathrm {~cm} ^ { 2 }\), is selected at random. Assuming that the number of daisies per \(\mathrm { cm } ^ { 2 }\) has a Poisson distribution,

1. find the probability that the chosen patch contains
   1. no daisies,
   2. one daisy. Ten such patches are chosen. Using your answers to part (a),
2. find the probability that the total number of daisies is less than two.
3. By considering the distribution of daisies over patches of \(10 \mathrm {~cm} ^ { 2 }\), use the Poisson distribution to find the probability that a particular area of \(10 \mathrm {~cm} ^ { 2 }\) contains no more than one daisy.
4. Compare your answers to parts (b) and (c).
5. Use a suitable approximation to find the probability that a patch of area \(1 \mathrm {~m} ^ { 2 }\) contains more than 8100 daisies.