Normally Distributed Apple Weights

00:00 7 marks

The weights, \(W\) grams, of apples harvested from a particular tree are normally distributed with a mean \(\mu\) and standard deviation \(\sigma\). It is known that 15% of the apples weigh less than 110 grams and 10% of the apples weigh more than 150 grams.

(a) Find the value of \(\mu\) and the value of \(\sigma\). [5]
(b) An apple is chosen at random. Find the probability that it weighs between 120 grams and 140 grams. [2]

Part a

7 minutes 5 marks

Question

Find the value of \(\mu\) and the value of \(\sigma\). [5]

Hint Hint

Hint

Use the inverse normal function for 15% and 90% probabilities to set up the equations \((110-\mu)/\sigma \approx -1.036\) and \((150-\mu)/\sigma \approx 1.282\).

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Complete Answer

\(\mu \approx 128\) grams and \(\sigma \approx 17.3\) grams

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Part b

5 minutes 2 marks

Question

An apple is chosen at random. Find the probability that it weighs between 120 grams and 140 grams. [2]

Hint Hint

Hint

Use the normal cumulative distribution function with \(\mu \approx 128\) and \(\sigma \approx 17.3\) to calculate \(P(120 < W < 140)\).

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Complete Answer

\(P(120 < W < 140) \approx 0.440\)

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