Edexcel S2 (Statistics 2)

1. Explain what is meant by
   1. a population,
   2. a sampling unit.
   Suggest suitable sampling frames for surveys of
2. families who have holidays in Greece,
3. mothers with children under two years old.

2. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k & 5 \leq x \leq 15 ,
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Find \(k\) and specify the cumulative density function \(\mathrm { F } ( x )\).
2. Write down the value of \(\mathrm { P } ( X < 8 )\).

3. A coin is tossed 20 times, giving 16 heads.

1. Test at the \(1 \%\) significance level whether the coin is fair, stating your hypotheses clearly.
2. Find the critical region for the same test at the \(0.1 \%\) significance level.

4. Alison and Gemma play table tennis. Alison starts by serving for the first five points. The probability that she wins a point when serving is \(p\).

1. Show that the probability that Alison is ahead at the end of her five serves is given by $$p ^ { 3 } \left( 6 p ^ { 2 } - 15 p + 10 \right) .$$
2. Evaluate this probability when \(p = 0.6\).

5. In a certain school, \(32 \%\) of Year 9 pupils are left-handed. A random sample of 10 Year 9 pupils is chosen.

1. Find the probability that none are left-handed.
2. Find the probability that at least two are left-handed.
3. Use a suitable approximation to find the probability of getting more than 5 but less than 15 left-handed pupils in a group of 35 randomly selected Year 9 pupils.
   Explain what adjustment is necessary when using this approximation. \section*{STATISTICS 2 (A) TEST PAPER 3 Page 2}

1. A sample of radioactive material decays randomly, with an approximate mean of 1.5 counts per minute.
   1. Name a distribution that would be suitable for modelling the number of counts per minute.
   Give any parameters required for the model.
2. Find the probability of at least 4 counts in a randomly chosen minute.
3. Find the probability of 3 counts or fewer in a random interval lasting 5 minutes. More careful measurements, over 50 one-minute intervals, give the following data for \(x\), the number of counts per minute: $$\sum x = 84 , \quad \sum x ^ { 2 } = 226$$
4. Decide whether these data support your answer to part (a).
5. Use the improved data to find probability of exactly two counts in a given one-minute interval.

7. Each day on the way to work, a commuter encounters a similar traffic jam. The length of time, in 10-minute units, spent waiting in the traffic jam is modelled by the random variable \(T\) with the cumulative distribution function: $$\begin{array} { l l } \mathrm { F } ( t ) = 0 & t < 0 ,
\mathrm {~F} ( t ) = \frac { t ^ { 2 } \left( 3 t ^ { 2 } - 16 t + 24 \right) } { 16 } & 0 \leq t \leq 2 ,
\mathrm {~F} ( t ) = 1 & t > 2 . \end{array}$$

1. Show that 0.77 is approximately the median value of \(T\).
2. Given that he has already waited for 12 minutes, find the probability that he will have to wait another 3 minutes.
3. Find, and sketch, the probability density function of \(T\).
4. Hence find the modal value of \(T\).
5. Comment on the validity of this model.