The thickness of brake pads produced by manufacturer $A$ is believed to be normally distributed. The thickness, in mm, of a random sample of 6 brake pads is recorded as follows: $$ \begin{array}{1111ll} 12.1 & 11.7 & 11.6 & 12.5 & 12.3 & 11.5 \end{array} $$ (i) Calculate unbiased estimates for the mean and the variance. Calculate a $95 \%$ confidence interval for the mean and a $95 \%$ confidence interval for the variance. (8 marks) (ii) A random sample of 10 brake pads made by another manufacturer, B, has mean thickness $11.7 \mathrm{--mm} $ and a standard deviation of $0.4 \mathrm{-mm} $. Test at $5 \%$ level of significance whether the brake pads made by manufacturer $A$ is on average thicker than those made by manufacturer B. Assume the thickness of brake pads made by the two manufacturers follow normal distribution with equal variance. SP.SS.3961