OCR S3 (Statistics 3) 2006 January

1 In order to judge the support for a new method of collecting household waste, a city council arranged a survey of 400 householders selected at random. The results showed that 186 householders were in favour of the new method.

1. Calculate a 95\% confidence interval for the proportion of all householders who are in favour of the new method. A city councillor said he believed that as many householders were in favour of the new method as were against it.
2. Comment on the councillor's belief.

2 A particular type of engine used in rockets is designed to have a mean lifetime of at least 2000 hours. A check of four randomly chosen engines yielded the following lifetimes in hours. $$\begin{array} { l l l l } 1896.4 & 2131.5 & 1903.3 & 1901.6 \end{array}$$ A significance test of whether engines meet the design is carried out. It may be assumed that lifetimes have a normal distribution.

1. Give a reason why a \(t\)-test should be used.
2. Carry out the test at the \(10 \%\) significance level.

3 For a restaurant with a home-delivery service, the delivery time in minutes can be modelled by a continuous random variable \(T\) with probability density function given by $$f ( t ) = \begin{cases} \frac { \pi } { 90 } \sin \left( \frac { \pi t } { 60 } \right) & 20 \leqslant t \leqslant 60
0 & \text { otherwise. } \end{cases}$$

1. Given that \(20 \leqslant a \leqslant 60\), show that \(\mathrm { P } ( T \leqslant a ) = \frac { 1 } { 3 } \left( 1 - 2 \cos \left( \frac { \pi a } { 60 } \right) \right)\). There is a delivery charge of \(\pounds 3\) but this is reduced to \(\pounds 2\) if the delivery time exceeds a minutes.
2. Find the value of \(a\) for which the expected value of the delivery charge for a home-delivery is £2.80.

4 A multi-storey car park has two entrances and one exit. During a morning period the numbers of cars using the two entrances are independent Poisson variables with means 2.3 and 3.2 per minute. The number leaving is an independent Poisson variable with mean 1.8 per minute. For a randomly chosen 10-minute period the total number of cars that enter and the number of cars that leave are denoted by the random variables \(X\) and \(Y\) respectively.

1. Use a suitable approximation to calculate \(\mathrm { P } ( X \geqslant 40 )\).
2. Calculate \(\mathrm { E } ( X - Y )\) and \(\operatorname { Var } ( X - Y )\).
3. State, giving a reason, whether \(X - Y\) has a Poisson distribution.

5 The continuous random variable \(X\) has cumulative distribution function given by $$F ( x ) = \begin{cases} 0 & x < 1 ,
\frac { 1 } { 8 } ( x - 1 ) ^ { 2 } & 1 \leqslant x < 3 ,
a ( x - 2 ) & 3 \leqslant x < 4 ,
1 & x \geqslant 4 , \end{cases}$$ where \(a\) is a positive constant.

1. Find the value of \(a\).
2. Verify that \(C _ { X } ( 8 )\), the 8th percentile of \(X\), is 1.8 .
3. Find the cumulative distribution function of \(Y\), where \(Y = \sqrt { X - 1 }\).
4. Find \(C _ { Y } ( 8 )\) and verify that \(C _ { Y } ( 8 ) = \sqrt { C _ { X } ( 8 ) - 1 }\).

6 A company with a large fleet of cars compared two types of tyres, \(A\) and \(B\). They measured the stopping distances of cars when travelling at a fixed speed on a dry road. They selected 20 cars at random from the fleet and divided them randomly into two groups of 10 , one group being fitted with tyres of type \(A\) and the other group with tyres of type \(B\). One of the cars fitted with tyres of type \(A\) broke down so these tyres were tested on only 9 cars. The stopping distances, \(x\) metres, for the two samples are summarised by $$n _ { A } = 9 , \quad \bar { x } _ { A } = 17.30 , \quad s _ { A } ^ { 2 } = 0.7400 , \quad n _ { B } = 10 , \quad \bar { x } _ { B } = 14.74 , \quad s _ { B } ^ { 2 } = 0.8160 ,$$ where \(s _ { A } ^ { 2 }\) and \(s _ { B } ^ { 2 }\) are unbiased estimates of the two population variances.
It is given that the two populations have the same variance.

1. Show that an unbiased estimate of this variance is 0.780 , correct to 3 decimal places. The population mean stopping distances for cars with tyres of types \(A\) and \(B\) are denoted by \(\mu _ { A }\) metres and \(\mu _ { B }\) metres respectively.
2. Stating any further assumption you need to make, calculate a \(98 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\). The manufacturers of Type \(B\) tyres assert that \(\mu _ { B } < \mu _ { A } - 2\).
3. Carry out a significance test of this assertion at the \(5 \%\) significance level. \section*{[Question 7 is printed overleaf.]}