Edexcel S2 (Statistics 2)

1. Explain briefly why it is often useful to take a sample from a population. [2 marks]
2. Suggest a suitable sampling frame for a local council to use to survey attitudes towards a proposed new shopping centre. [1 mark]

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. [6 marks]

A random variable \(X\) has a Poisson distribution with a mean, \(\lambda\), which is assumed to equal 5.

1. Find P\((X = 0)\). [1 mark]
2. 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 \(\lambda < 5\). [8 marks]

The waiting time, in minutes, at a dentist is modelled by the continuous random variable \(T\) with probability density function $$f(t) = k(10 - t) \quad 0 \leq t \leq 10,$$ $$f(t) = 0 \quad \text{otherwise}.$$

1. Sketch the graph of \(f(t)\) and find the value of \(k\). [4 marks]
2. Find the mean value of \(T\). [4 marks]
3. Find the 95th percentile of \(T\). [3 marks]
4. 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. [2 marks]

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. [1 mark]
2. Find the probability that a particular page has more than 2 misprints. [3 marks]
3. Find the probability that Chapter 1, with 8 pages, has no misprints at all. [2 marks]

Chapter 2 is longer, at 20 pages.

1. 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. [7 marks]

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 2.4.

1. Show that \(n = 15\). [2 marks]
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, [3 marks]
   2. at least 5. [2 marks]

Ten samples of 15 bags each are tested. Find the probability that

1. all these batches contain less than 5 underweight bags, [3 marks]
2. the fourth batch tested is the first to contain less than 5 underweight bags. [3 marks]

A continuous random variable \(X\) has a probability density function given by $$f(x) = \frac{x^2}{312} \quad 4 \leq x \leq 10,$$ $$f(x) = 0 \quad \text{otherwise}.$$

1. Find E\((X)\). [3 marks]
2. Find the variance of \(X\). [4 marks]
3. Find the cumulative distribution function F\((x)\), for all values of \(x\). [5 marks]
4. Hence find the median value of \(X\). [3 marks]
5. Write down the modal value of \(X\). [1 mark]

It is sometimes suggested that, for most distributions, $$2 \times (\text{median} - \text{mean}) \approx \text{mode} - \text{median}.$$

1. Show that this result is not satisfied in this case, and suggest a reason why. [2 marks]