2010 HCI P2 Q9

IB AA SL and HL: core and optional content

Core: apple mean and binomial count. Optional: comparison of total orange payments.

An orchard owned by Mark produces oranges which have masses (in grams) that follow a normal distribution \(\text{N}\hspace{0.1em}\left( 190,576 \right)\). Visitors to this orchard can buy the oranges at \(\$0.10\) per \(100\) g.

Find the probability that the payment made by a visitor buying \(20\) oranges will differ from \(2\) times the payment made by another visitor buying \(10\) oranges by at most \(\$0.15\). [4]

The orchard produces apples which are graded according to their mass. Apples with a mass exceeding \(150\) g are graded as ‘large’ while apples with a mass less than \(70\) g are graded as ‘small’.

Mark finds that the proportion of apples graded as ‘large’ is the same as that of the apples graded as ‘small’. It is given that the mass of a randomly chosen apple from Mark’s orchard follows a normal distribution with mean \(\mu \) g and standard deviation \(30\) g.

1. Find the value of \(\mu \). [1]
2. What is the probability that Mark will get at least \(5\) apples graded as ‘large’ when he randomly selects \(65\) apples from his orchard? [3]

Modelling assumption for this exercise: masses of distinct selected fruits are mutually independent and follow the stated distribution for their type.