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.
- 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.
- 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 \(\mu _ { B } < \mu _ { A } - 2\).
- Carry out a significance test of this assertion at the \(5 \%\) significance level.
\section*{[Question 7 is printed overleaf.]}