Edexcel S2 (Statistics 2) 2006 January

1. A fair coin is tossed 4 times.

Find the probability that

1. an equal number of head and tails occur
2. all the outcomes are the same,
3. the first tail occurs on the third throw.

2. Accidents on a particular stretch of motorway occur at an average rate of 1.5 per week.

1. Write down a suitable model to represent the number of accidents per week on this stretch of motorway. Find the probability that
2. there will be 2 accidents in the same week,
3. there is at least one accident per week for 3 consecutive weeks,
4. there are more than 4 accidents in a 2 week period.

3. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).

1. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\). Find
2. \(\mathrm { E } ( X )\),
3. \(\operatorname { Var } ( \mathrm { X } )\),
4. \(\mathrm { P } ( - 0.3 < X < 3.3 )\).

4. The random variable \(X \sim \mathrm {~B} ( 150,0.02 )\). Use a suitable approximation to estimate \(\mathrm { P } ( X > 7 )\).

5. A continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) where $$f ( x ) = \begin{cases} k x ( x - 2 ) , & 2 \leq x \leq 3
0 , & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Show that \(k = \frac { 3 } { 4 }\). Find
2. \(\mathrm { E } ( X )\),
3. the cumulative distribution function \(\mathrm { F } ( x )\).
4. Show that the median value of \(X\) lies between 2.70 and 2.75.

6. A bag contains a large number of coins. Half of them are 1 p coins, one third are 2 p coins and the remainder are 5p coins.

1. Find the mean and variance of the value of the coins. A random sample of 2 coins is chosen from the bag.
2. List all the possible samples that can be drawn.
3. Find the sampling distribution of the mean value of these samples.

7. A teacher thinks that \(20 \%\) of the pupils in a school read the Deano comic regularly. He chooses 20 pupils at random and finds 9 of them read the Deano.

1. 1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the percentage of pupils that read the Deano is different from 20\%. State your hypotheses clearly.
   2. State all the possible numbers of pupils that read the Deano from a sample of size 20 that will make the test in part (a)(i) significant at the \(5 \%\) level.
      (9) The teacher takes another 4 random samples of size 20 and they contain 1, 3, 1 and 4 pupils that read the Deano.
2. By combining all 5 samples and using a suitable approximation test, at the \(5 \%\) level of significance, whether or not this provides evidence that the percentage of pupils in the school that read the Deano is different from 20\%.
3. Comment on your results for the tests in part (a) and part (b).