OCR S2 (Statistics 2) 2011 June

1 In Fisher Avenue there are 263 houses, numbered 1 to 263. Explain how to obtain a random sample of 20 of these houses.

2 The random variable \(Y\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that $$\mathrm { P } ( Y < 48.0 ) = \mathrm { P } ( Y > 57.0 ) = 0.0668 .$$ Find the value \(y _ { 0 }\) such that \(\mathrm { P } \left( Y > y _ { 0 } \right) = 0.05\).

3 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 5 ^ { 2 } \right)\). A hypothesis test is carried out of \(\mathrm { H } _ { 0 } : \mu = 20.0\) against \(\mathrm { H } _ { 1 } : \mu < 20.0\), at the \(1 \%\) level of significance, based on the mean of a sample of size 16. Given that in fact \(\mu = 15.0\), find the probability that the test results in a Type II error.

4 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 } { 16 } ( x - 2 ) ^ { 2 } & 0 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. Calculate the variance of \(X\).
3. A student writes " \(X\) is more likely to occur when \(x\) takes values further away from 2 ". Explain whether you agree with this statement.

5 A travel company finds from its records that \(40 \%\) of its customers book with travel agents. The company redesigns its website, and then carries out a survey of 10 randomly chosen customers. The result of the survey is that 1 of these customers booked with a travel agent.

1. Test at the \(5 \%\) significance level whether the percentage of customers who book with travel agents has decreased.
2. The managing director says that "Our redesigned website has resulted in a decrease in the percentage of our customers who book with travel agents." Comment on this statement.

6 Records show that before the year 1990 the maximum daily temperature \(T ^ { \circ } \mathrm { C }\) at a seaside resort in August can be modelled by a distribution with mean 24.3. The maximum temperatures of a random sample of 50 August days since 1990 can be summarised by $$n = 50 , \quad \Sigma t = 1314.0 , \quad \Sigma t ^ { 2 } = 36602.17 .$$

1. Test, at the \(1 \%\) significance level, whether there is evidence of a change in the mean maximum daily temperature in August since 1990.
2. Give a reason why it is possible to use the Central Limit Theorem in your test.

7 The number of customer complaints received by a company per day is denoted by \(X\). Assume that \(X\) has the distribution \(\operatorname { Po } ( 2.2 )\).

1. In a week of 5 working days, the probability there are at least \(n\) customer complaints is 0.146 correct to 3 significant figures. Use tables to find the value of \(n\).
2. Use a suitable approximation to find the probability that in a period of 20 working days there are fewer than 38 customer complaints. A week of 5 working days in which at least \(n\) customer complaints are received, where \(n\) is the value found in part (i), is called a 'bad' week.
3. Use a suitable approximation to find the probability that, in 40 randomly chosen weeks, more than 7 are bad.

8

1. A group of students is discussing the conditions that are needed if a Poisson distribution is to be a good model for the number of telephone calls received by a fire brigade on a working day.
   1. Alice says "Events must be independent". Explain why this condition may not hold in this context.
   2. State a different condition that is needed. Explain whether it is likely to hold in this context.
2. The random variables \(R , S\) and \(T\) have independent Poisson distributions with means \(\lambda , \mu\) and \(\lambda + \mu\) respectively.
   1. In the case \(\lambda = 2.74\), find \(\mathrm { P } ( R > 2 )\).
   2. In the case \(\lambda = 2\) and \(\mu = 3\), find \(\mathrm { P } ( R = 0\) and \(S = 1 ) + \mathrm { P } ( R = 1\) and \(S = 0 )\). Give your answer correct to 4 decimal places.
   3. In the general case, show algebraically that $$\mathrm { P } ( R = 0 \text { and } S = 1 ) + \mathrm { P } ( R = 1 \text { and } S = 0 ) = \mathrm { P } ( T = 1 ) .$$