Edexcel S2 (Statistics 2) 2002 June

A random sample \(X_1, X_2, \ldots, X_n\) is taken from a finite population. A statistic \(Y\) is based on this sample.

1. Explain what you understand by the statistic \(Y\). [2]
2. Give an example of a statistic. [1]
3. Explain what you understand by the sampling distribution of \(Y\). [2]

The continuous random variable \(R\) is uniformly distributed on the interval \(\alpha \leq R \leq \beta\). Given that \(\mathrm{E}(R) = 3\) and \(\mathrm{Var}(R) = \frac{25}{3}\), find

1. the value of \(\alpha\) and the value of \(\beta\), [7]
2. \(\mathrm{P}(R < 6.6)\). [2]

Past records show that 20\% of customers who buy crisps from a large supermarket buy them in single packets. During a particular day a random sample of 25 customers who had bought crisps was taken and 2 of them had bought them in single packets.

1. Use these data to test, at the 5\% level of significance, whether or not the percentage of customers who bought crisps in single packets that day was lower than usual. State your hypotheses clearly. [6]

At the same supermarket, the manager thinks that the probability of a customer buying a bumper pack of crisps is 0.03. To test whether or not this hypothesis is true the manager decides to take a random sample of 300 customers.

1. Stating your hypotheses clearly, find the critical region to enable the manager to test whether or not there is evidence that the probability is different from 0.03. The probability for each tail of the region should be as close as possible to 2.5\%. [6]
2. Write down the significance level of this test. [1]

A garden centre sells canes of nominal length 150 cm. The canes are bought from a supplier who uses a machine to cut canes of length \(L\) where \(L \sim \mathrm{N}(\mu, 0.3^2)\).

1. Find the value of \(\mu\), to the nearest 0.1 cm, such that there is only a 5\% chance that a cane supplied to the garden centre will have length less than 150 cm. [4]

A customer buys 10 of these canes from the garden centre.

1. Find the probability that at most 2 of the canes have length less than 150 cm. [3]

Another customer buys 500 canes.

1. Using a suitable approximation, find the probability that fewer than 35 of the canes will have length less than 150 cm. [6]

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{x}{15}, & 0 \leq x \leq 2, \\ \frac{2}{15}, & 2 < x < 7, \\ \frac{4}{9} - \frac{2x}{45}, & 7 \leq x \leq 10, \\ 0, & \text{otherwise}. \end{cases}$$

1. Sketch \(f(x)\) for all values of \(x\). [3]
2. 1. Find expressions for the cumulative distribution function, \(\mathrm{F}(x)\), for \(0 \leq x \leq 2\) and for \(7 \leq x \leq 10\).
   2. Show that for \(2 < x < 7\), \(\mathrm{F}(x) = \frac{2x}{15} - \frac{2}{15}\).
   3. Specify \(\mathrm{F}(x)\) for \(x < 0\) and for \(x > 10\).
   [8]
3. Find \(\mathrm{P}(X \leq 8.2)\). [2]
4. Find, to 3 significant figures, \(\mathrm{E}(X)\). [4]