Edexcel S2 (Statistics 2)

1. A random sample is to be taken from the A-level results obtained by the final-year students in a Sixth Form College. Suggest
   1. suitable sampling units,
   2. a suitable sampling frame.
   3. Would it be advisable simply to use the results of all those doing A-level Maths?
   Explain your answer.

2. The random variable \(X\), which can take any value in the interval \(1 \leq X \leq n\), is modelled by the continuous uniform distribution with mean 12.

1. Show that \(n = 23\) and find the variance of \(X\).
2. Find \(\mathrm { P } ( 10 < X < 14 )\).

3. The Driving Theory Test includes 30 questions which require one answer to be selected from four options.

1. Phil ticks answers at random. Find how many of the 30 he should expect to get right.
2. If he gets 15 correct, decide whether this is evidence that he has actually done some revision. Use a \(5 \%\) significance level. Another candidate, Sarah, has revised and has a 0.9 probability of getting each question right.
3. Determine the expected number of answers that Sarah will get right.
4. Find the probability that Sarah gets more than 25 correct answers out of 30.

4. A continuous random variable \(X\) has probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 1 ,
\mathrm { f } ( x ) = k x & 1 \leq x \leq 4 ,
\mathrm { f } ( x ) = 0 & x > 4 . \end{array}$$

1. Sketch a graph of \(\mathrm { f } ( x )\), and hence find the value of \(k\).
2. Calculate the mean and the variance of \(X\). \section*{STATISTICS 2 (A)TEST PAPER 4 Page 2}

1. In World War II, the number of V2 missiles that landed on each square mile of London was, on average, \(3 \cdot 5\). Assuming that the hits were randomly distributed throughout London,
   1. suggest a suitable model for the number of hits on each square mile, giving a suitable value for any parameters.
   2. calculate the probability that a particular square mile received
      1. no hits,
      2. more than 7 hits.
   3. State, with a reason, whether the model is likely to be accurate.
   In contrast, the number of bombs weighing more than 1 ton landing on each square mile was 45 .
2. Use a suitable approximation to find the probability that a randomly selected square mile received more than 60 such bombs. Explain what adjustment must be made when using this approximation.

6. In a fruit packing plant, apples are packed on to trays of 10 , and then checked for blemishes. The chance of any particular apple having a blemish is \(5 \%\). If a tray is selected at random, find

1. the probability that at least two of the apples in it are blemished,
2. the probability that exactly two are blemished. Trays are now packed in boxes of 50 trays each. In one such box, find
3. the probability that at most one tray contains at least two blemished apples,
4. the expected number of trays containing at least two blemished apples.
5. Use a suitable approximation to find the probability that in a random selection of 20 trays there are more than 10 blemished apples.

7. The time, in hours, taken to run the London marathon is modelled by a continuous random variable \(T\) with the probability density function $$f ( t ) = \begin{cases} c ( t - 2 ) & 2 \leq t < 4
\frac { 2 c ( 7 - t ) } { 3 } & 4 \leq t \leq 7
0 & \text { otherwise } \end{cases}$$

1. Sketch the function \(\mathrm { f } ( t )\), and show that \(c = \frac { 1 } { 5 }\).
2. Calculate the median value of \(T\).
3. Make two critical comments about the model.