Edexcel S2 (Statistics 2)

A company that makes ropes for mountaineering wants to assess the breaking strain of its ropes.

1. Explain why a sample survey, and not a census, should be used. [2 marks]
2. Suggest an appropriate sampling frame. [1 mark]

It is thought that a random variable \(X\) has a Poisson distribution whose mean, \(\lambda\), is equal to 8. Find the critical region to test the hypothesis \(H_0 : \lambda = 8\) against the hypothesis \(H_1 : \lambda < 8\), working at the 1\% significance level. [5 marks]

A child cuts a 30 cm piece of string into two parts, cutting at a random point.

1. Name the distribution of \(L\), the length of the longer part of string, and sketch the probability density function for \(L\). [4 marks]
2. Find the probability that one part of the string is more than twice as long as the other. [3 marks]

A supplier of widgets claims that only 10\% of his widgets have faults.

1. In a consignment of 50 widgets, 9 are faulty. Test, at the 5\% significance level, whether this suggests that the supplier's claim is false. [6 marks]
2. Find how many faulty widgets would be needed to provide evidence against the claim at the 1\% significance level. [3 marks]

In a survey of 22 families, the number of children, \(X\), in each family was given by the following table, where \(f\) denotes the frequency:

\(X\)	0	1	2	3	4	5
\(f\)	3	8	5	3	2	1

1. Find the mean and variance of \(X\). [4 marks]
2. Explain why these results suggest that \(X\) may follow a Poisson distribution. [1 mark]
3. State another feature of the data that suggests a Poisson distribution. [1 mark]

It is sometimes suggested that the number of children in a family follows a Poisson distribution with mean 2·4. Assuming that this is correct,

1. find the probability that a family has less than two children. [3 marks]
2. Use this result to find the probability that, in a random sample of 22 families, exactly 11 of the families have less than two children. [3 marks]

When a park is redeveloped, it is claimed that 70\% of the local population approve of the new design. Assuming this to be true, find the probability that, in a group of 10 residents selected at random,

1. 6 or more approve, [3 marks]
2. exactly 7 approve. [3 marks]

A conservation group, however, carries out a survey of 20 people, and finds that only 9 approve.

1. Use this information to carry out a hypothesis test on the original claim, working at the 5\% significance level. State your conclusion clearly. [5 marks]

If the conservationists are right, and only 45\% approve of the new park,

1. use a suitable approximation to the binomial distribution to estimate the probability that in a larger survey, of 500 people, less than half will approve. [7 marks]

A continuous random variable \(X\) has probability density function f(x) given by $$\text{f(x)} = \frac{2x}{3} \quad 0 \leq x < 1,$$ $$\text{f(x)} = 1 - \frac{x}{3} \quad 1 \leq x \leq 3,$$ $$\text{f(x)} = 0 \quad \text{otherwise}.$$

1. Sketch the graph of f(x) for all \(x\). [3 marks]
2. Find the mean of \(X\). [5 marks]
3. Find the standard deviation of \(X\). [7 marks]
4. Show that the cumulative distribution function of \(X\) is given by $$\text{F(x)} = \frac{x^2}{3} \quad 0 \leq x < 1,$$ and find F(x) for \(1 \leq x \leq 3\). [6 marks]