OCR S2 (Statistics 2) 2009 June

1 The random variable \(H\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). It is given that \(\mathrm { P } ( H < 105.0 ) = 0.2420\) and \(\mathrm { P } ( H > 110.0 ) = 0.6915\). Find the values of \(\mu\) and \(\sigma\), giving your answers to a suitable degree of accuracy.

2 The random variable \(D\) has the distribution \(\operatorname { Po } ( 20 )\). Using an appropriate approximation, which should be justified, calculate \(\mathrm { P } ( D \geqslant 25 )\).

3 An electronics company is developing a new sound system. The company claims that \(60 \%\) of potential buyers think that the system would be good value for money. In a random sample of 12 potential buyers, 4 thought that it would be good value for money. Test, at the 5\% significance level, whether the proportion claimed by the company is too high.

4 A survey is to be carried out to draw conclusions about the proportion \(p\) of residents of a town who support the building of a new supermarket. It is proposed to carry out the survey by interviewing a large number of people in the high street of the town, which attracts a large number of tourists.

1. Give two different reasons why this proposed method is inappropriate.
2. Suggest a good method of carrying out the survey.
3. State two statistical properties of your survey method that would enable reliable conclusions about \(p\) to be drawn.

5 In a large region of derelict land, bricks are found scattered in the earth.

1. State two conditions needed for the number of bricks per cubic metre to be modelled by a Poisson distribution. Assume now that the number of bricks in 1 cubic metre of earth can be modelled by the distribution Po(3).
2. Find the probability that the number of bricks in 4 cubic metres of earth is between 8 and 14 inclusive.
3. Find the size of the largest volume of earth for which the probability that no bricks are found is at least 0.4.

6 The continuous random variable \(R\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The results of 100 observations of \(R\) are summarised by $$\Sigma r = 3360.0 , \quad \Sigma r ^ { 2 } = 115782.84 .$$

1. Calculate an unbiased estimate of \(\mu\) and an unbiased estimate of \(\sigma ^ { 2 }\).
2. The mean of 9 observations of \(R\) is denoted by \(\bar { R }\). Calculate an estimate of \(\mathrm { P } ( \bar { R } > 32.0 )\).
3. Explain whether you need to use the Central Limit Theorem in your answer to part (ii).

7 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { 9 } x ( 3 - x ) & 0 \leqslant x \leqslant 3 ,
0 & \text { otherwise } . \end{cases}$$

1. Find the variance of \(X\).
2. Show that the probability that a single observation of \(X\) lies between 0.0 and 0.5 is \(\frac { 2 } { 27 }\).
3. 108 observations of \(X\) are obtained. Using a suitable approximation, find the probability that at least 10 of the observations lie between 0.0 and 0.5 .
4. The mean of 108 observations of \(X\) is denoted by \(\bar { X }\). Write down the approximate distribution of \(\bar { X }\), giving the value(s) of any parameter(s).

8 In a large company the time taken for an employee to carry out a certain task is a normally distributed random variable with mean 78.0 s and unknown variance. A new training scheme is introduced and after its introduction the times taken by a random sample of 120 employees are recorded. The mean time for the sample is 76.4 s and an unbiased estimate of the population variance is \(68.9 \mathrm {~s} ^ { 2 }\).

1. Test, at the \(1 \%\) significance level, whether the mean time taken for the task has changed.
2. It is required to redesign the test so that the probability of making a Type I error is less than 0.01 when the sample mean is 77.0 s . Calculate an estimate of the smallest sample size needed, and explain why your answer is only an estimate.