Edexcel S2 (Statistics 2)

1. (a) Explain briefly why it is often useful to take a sample from a population.
   (b) Suggest a suitable sampling frame for a local council to use to survey attitudes towards a proposed new shopping centre.
2. A certain type of lettuce seed has a \(12 \%\) failure rate for germination. In a new sample of 25 genetically modified seeds, only 1 did not germinate.
   Clearly stating your hypotheses, test, at the \(1 \%\) significance level, whether the GM seeds are better.
3. A random variable \(X\) has a Poisson distribution with a mean, \(\lambda\), which is assumed to equal 5 .
   (a) Find \(\mathrm { P } ( X = 0 )\).
   (b) In 100 measurements, the value 0 occurs three times. Find the highest significance level at which you should reject the original hypothesis in favour of \(\lambda < 5\).
4. The waiting time, in minutes, at a dentist is modelled by the continuous random variable \(T\) with probability density function

$$\begin{array} { l l } \mathrm { f } ( t ) = k ( 10 - t ) & 0 \leq t \leq 10
\mathrm { f } ( t ) = 0 & \text { otherwise. } \end{array}$$ (a) Sketch the graph of \(\mathrm { f } ( t )\) and find the value of \(k\).
(b) Find the mean value of \(T\).
(c) Find the 95th percentile of \(T\).
(d) State whether you consider this function to be a sensible model for \(T\) and suggest how it could be modified to provide a better model.

5. A textbook contains, on average, 1.2 misprints per page. Assuming that the misprints are randomly distributed throughout the book,

1. specify a suitable model for \(X\), the random variable representing the number of misprints on a given page.
2. Find the probability that a particular page has more than 2 misprints.
3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. Chapter 2 is longer, at 20 pages.
4. Use a suitable approximation to find the probability that Chapter 2 has less than ten misprints altogether. Explain what adjustment is necessary when making this approximation. \section*{STATISTICS 2 (A) TEST PAPER 5 Page 2}

1. On a production line, bags are filled with cement and weighed as they emerge. It is found that \(20 \%\) of the bags are underweight. In a random sample consisting of \(n\) bags, the variance of the number of underweight bags is found to be \(2 \cdot 4\).
   1. Show that \(n = 15\).
   2. Use cumulative binomial probability tables to find the probability that, in a further random sample of 15 bags, the number that are underweight is
      1. less than 3 ,
      2. at least 5 .
   Ten samples of 15 bags each are tested. Find the probability that
2. all these batches contain less than 5 underweight bags,
3. the fourth batch tested is the first to contain less than 5 underweight bags.

7. A continuous random variable \(X\) has a probability density function given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { x ^ { 2 } } { 312 } & 4 \leq x \leq 10
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Find \(\mathrm { E } ( X )\).
2. Find the variance of \(X\).
3. Find the cumulative distribution function \(\mathrm { F } ( x )\), for all values of \(x\).
4. Hence find the median value of \(X\).
5. Write down the modal value of \(X\). It is sometimes suggested that, for most distributions, $$2 \times ( \text { median } - \text { mean } ) \approx \text { mode } - \text { median } .$$
6. Show that this result is not satisfied in this case, and suggest a reason why.