Edexcel S2 (Statistics 2)

1. (a) Explain the difference between a discrete and a continuous variable.

A random number generator on a calculator generates numbers, \(X\), to 3 decimal places, in the range 0 to 1 , e.g. 0.386 . The variable \(X\) may be modelled by a continuous uniform distribution, having the probability density function \(\mathrm { f } ( x )\), where $$\begin{array} { l l } \mathrm { f } ( x ) = 1 & 0 < x < 1
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$ (b) Explain why this model is not totally accurate.
(c) Sketch the cumulative distribution function of \(X\).

2. A video rental shop needs to find out whether or not videos have been rewound when they are returned; it will do this by taking a sample of returned videos

1. State one advantage and one disadvantage of taking a sample.
2. Suggest a suitable sampling frame.
3. Describe the sampling units.
4. Criticise the sampling method of looking at just one particular shelf of videos.

3. The random variable \(X\) is modelled by a binomial distribution \(\mathrm { B } ( n , p )\), with \(n = 20\) and \(p\) unknown. It is suspected that \(p = 0 \cdot 4\).

1. Find the critical region for the test of \(\mathrm { H } _ { 0 } : p = 0.4\) against \(\mathrm { H } _ { 1 } : p \neq 0.4\), at the \(5 \%\) significance level.
2. Find the critical region if, instead, the alternative hypothesis is \(\mathrm { H } _ { 1 } : p < 0.4\).

4. A random variable \(X\) has the distribution \(\mathrm { B } ( 80,0.375 )\).

1. Write down the mean and variance of \(X\).
2. Use the Normal approximation to the binomial distribution to estimate \(\mathrm { P } ( X > 40 )\).

5. A traffic analyst is interested in the number of heavy lorries passing a certain junction. He counts the numbers of lorries in 100 five-minute intervals, and gets the following results: Q. 5 continued on next page ... \section*{STATISTICS 2 (A) TEST PAPER 9 Page 2}

1. continued ...
   1. Show that the mean of \(X\) is 3 , and find the variance of \(X\).
   2. Give two reasons for thinking that \(X\) can be modelled by a Poisson distribution. (2 marks)
   After a new landfill site has been established nearby, a member of an environmental group notices that 18 lorries pass the junction in a period of 15 minutes. The group claims that this is evidence that the mean number of lorries per five-minute interval has increased.
2. Test whether the group's claim is valid. Work at the \(5 \%\) significance level, and state your hypotheses clearly.

6. In a particular parliamentary constituency, the percentage of Conservative voters at the last election was \(35 \%\), and the percentage who voted for the Monster Raving Loony party was \(2 \%\).

1. Find the probability that a random sample of 10 electors includes at least two Conservative voters. Use suitable approximations to find
2. the probability that a random sample of 500 electors will include at least 200 who voted either Conservative or Monster Raving Loony,
3. the probability that a random sample of 200 electors will have at least 5 Monster Raving Loony voters in it.
4. One of (b) or (c) requires an adjustment to be made before a calculation is done. Explain what this adjustment is, and why it is necessary.

7. The fraction of sky covered by cloud is modelled by the random variable \(X\) with probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = 0 & x < 0
\mathrm { f } ( x ) = k x ^ { 2 } ( 1 - x ) & 0 \leq x \leq 1 ,
\mathrm { f } ( x ) = 0 & x > 1 . \end{array}$$

1. Find \(k\) and sketch the graph of \(\mathrm { f } ( x )\).
2. Find the mean and the variance of \(X\).
3. Find the cumulative distribution function \(\mathrm { F } ( x )\).
4. Given that flying is prohibited when \(85 \%\) of the sky is covered by cloud, show that cloud conditions allow flying nearly \(90 \%\) of the time.