All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)

Let X, Y be the output daily from mines A and B, respectively.

$latex \mathrm{P}(X<75) = 0.9811$ $latex \Rightarrow \mathrm{P}(Z< \frac{25}{\sigma}) = 0.9811$ $latex \frac{25}{\sigma} = 2.07701689$ $latex \sigma^{2} = 144.88 \approx 145$ (ii) $latex Y ~\sim~ \mathrm{N}(75, 64)$ $latex Y_1 + \ldots +Y_7 ~\sim~ \mathrm{N}(525,448)$ $latex \mathrm{P}(Y_1 + \ldots +Y_7 < 500) = 0.119$ (iii) $latex Y_1 + \ldots + Y_7 - 2(X_1 + \ldots +X_5) ~\sim~ \mathrm{N} (25, 3348)$ $latex \mathrm{P}[Y_1 + \ldots + Y_7 - 2(X_1 + \ldots +X_5) > 0] = 0.667$

### KS Comments

Some students forgot to change the (i)’s probability before applying inverse normal. This proved to be costly as the whole question was ruined. Some students applied the expectations and variance wrongly too.