2018 Jan

Question 5

(a) Cheese is sold in blocks of two different sizes, large and small. The weight of a

randomly chosen small block, X, is normally distributed with mean 253 grams

and standard deviation 4 grams. The weight of a randomly chosen large block,

Y, is normally distributed with mean 505 grams and standard deviation 6 grams.

The weight of any block of cheese may be assumed to be independent of the

weight of any other blocks.

Determine m such that P(X>m) = 0.894.

If 3 small blocks of cheese are randomly chosen, compute the

probability that one of them weighs more than 253 grams and each of

the remaining two weighs between 245 and 253 grams.

(iii) Compute the probability that the difference between the total weight of 2

randomly chosen large blocks and 4 times the weight of a randomly selected

small block is at least 5 grams

(iv) Due to a change in the packaging process, the mean weight of a

randomly chosen large block of cheese is changed to grams. If the

standard deviation is still 6 grams, determine the greatest integer value

of such that less than 90 % of the large blocks produced weigh more

than 500 grams.