Edexcel S2 (Statistics 2)

Briefly explain what is meant by

1. a statistical model, [2 marks]
2. a sampling frame, [1 mark]
3. a sampling unit. [1 mark]

1. Explain what is meant by the critical region of a statistical test. [2 marks]
2. Under a hypothesis \(H_0\), an event \(A\) can happen with probability \(4 \cdot 2\%\). The event \(A\) does then happen. State, with justification, whether \(H_0\) should be accepted or rejected at the \(5\%\) significance level. [2 marks]

1. Briefly describe the main features of a binomial distribution. [2 marks]

I conduct an experiment by randomly selecting 10 cards, without replacement, from a normal pack of 52.

1. Explain why the distribution of \(X\), the number of hearts obtained, is not \(\text{B}(10, \frac{1}{4})\). [2 marks]

After making the appropriate adjustment to the experiment, which should be stated, so that the distribution is \(\text{B}(10, \frac{1}{4})\), find

1. the probability of getting no hearts, [3 marks]
2. the probability of getting 4 or more hearts. [2 marks]
3. If the modified experiment is repeated 50 times, find the total number of hearts that you would you expect to have drawn. [2 marks]

A Geiger counter is observed in the presence of a radioactive source. In 100 one-minute intervals, the number of counts recorded are as follows:

No of counts, \(X\)	0	1	2	3	4	5	6
Frequency	10	24	29	16	12	6	3

1. Find the mean and variance of this data, and show that it supports the idea that the random variable \(X\) is following a Poisson distribution. [5 marks]
2. Use a Poisson distribution with the mean found in part (a) to calculate, to 3 decimal places, the probability that more than 6 counts will be recorded in any particular minute. [4 marks]
3. Find the number of one-minute intervals, in the sample of 100, in which more than 6 counts would be expected. [2 marks]

A continuous random variable \(X\) has the cumulative distribution function $$F(x) = 0 \quad x < 2,$$ $$F(x) = k(x - a)^2 \quad 2 \leq x \leq 6,$$ $$F(x) = 1 \quad x \geq 6.$$

1. Find the values of the constants \(a\) and \(k\). [4 marks]
2. Show that the median of \(X\) is \(2(1 + \sqrt{2})\). [4 marks]
3. Given that \(X > 4\), find the probability that \(X > 5\). [6 marks]

A small opinion poll shows that the Trendies have a \(10\%\) lead over the Oldies. The poll is based on a survey of 20 voters, in which the Trendies got 11 and the Oldies 9. The Oldies spokesman says that the result is consistent with a \(10\%\) lead for the Oldies, whilst the Trendies spokesperson says that this is impossible.

1. At the \(5\%\) significance level, test which is right, stating your null hypothesis carefully. [6 marks]
2. If it is indeed true that the Trendies are supported by \(55\%\) of the population, use a suitable approximation to find the probability that in a random sample of 200 voters they would obtain less than half of the votes. [8 marks]

A continuous random variable \(X\) has the probability density function $$\text{f}(x) = \frac{6x}{175} \quad 0 \leq x < 5,$$ $$\text{f}(x) = \frac{6x(10-x)}{875} \quad 5 \leq x \leq 10,$$ $$\text{f}(x) = 0 \quad \text{otherwise}.$$

1. Verify that f is a probability density function. [6 marks]
2. Write down the probability that \(X < 1\). [2 marks]
3. Find the cumulative distribution function of \(X\), carefully showing how it changes for different domains. [7 marks]
4. Find the probability that \(2 < X < 7\). [2 marks]