6. Brickland and Goodbrick are two manufacturers of bricks. The lengths of the bricks produced by each manufacturer can be assumed to be normally distributed. A random sample of 20 bricks is taken from Brickland and the length, \(x \mathrm {~mm}\), of each brick is recorded. The mean of this sample is 207.1 mm and the variance is \(3.2 \mathrm {~mm} ^ { 2 }\).

1. Calculate the \(98 \%\) confidence interval for the mean length of brick from Brickland. A random sample of 10 bricks is selected from those manufactured by Goodbrick. The length of each brick, \(y \mathrm {~mm}\), is recorded. The results are summarised as follows. $$\sum y = 2046.2 \quad \sum y ^ { 2 } = 418785.4$$ The variances of the length of brick for each manufacturer are assumed to be the same.
2. Find a \(90 \%\) confidence interval for the value by which the mean length of brick made by Brickland exceeds the mean length of brick made by Goodbrick.