10

10. [Maximum mark: 17]

At Adam’s Apple Orchard the weights of apples, $W$, in grams, are normally distributed with a mean 175 grams and standard deviation 8 grams.

(a) Find the probability that a randomly chosen apple weighs less than 170 grams. [2]

(b) It is found that 20% of the apples weigh more than $w$ grams. Find $w$, correct to four significant figures. [2]

All orchards classify an apple as premium when its weight is between 170 and 185 grams.

(c) Find the percentage of apples that are classified as premium at Adam’s Apple Orchard. [2]

After orders are completed, there are many apples left over. Boxes are filled with randomly chosen left‑over apples. Each box contains 40 apples.

(d) Find the probability that a randomly chosen box contains at least 30 premium apples. [3]

(e) If 10 of these boxes are randomly selected, find the probability that exactly 4 boxes have at least 30 premium apples. [2]

At a neighbouring orchard the weights of apples, $M$, in grams, are normally distributed with mean $\mu$ and standard deviation $\sigma$. It is known that:
• 82% of their apples are classified as premium
• the percentage of apples that weigh less than 170 grams is twice the percentage of apples that weigh more than 185 grams.

(f) Find the value of $\mu$. [6]