5. [Maximum mark: 17]
The weights of apples harvested from an orchard are normally distributed with a mean of 160 g and a standard deviation of 15 g. Apples weighing less than 140 g are classified as small.
(a) Find the probability that a randomly selected apple is classified as small. [2]
Apples weighing between 140 g and 175 g are classified as medium.
(b) Find the probability that a randomly selected apple is classified as medium. [2]
(c) Out of a sample of 200 randomly selected apples, find the expected number of apples classified as medium. [2]
Apples weighing more than w grams are classified as large. 10% of apples are classified as large.
(d) Find the value of w. [3]
A supermarket orders 500 apples.
(e) Find the probability that at least 100 of these apples are classified as small. [3]
It is known that 5% of all apples harvested have blemishes. An apple is selected at random.
(f) Find the probability that the selected apple is small and has blemishes, assuming weight and blemishes are independent events. [2]
(g) Given that a selected apple is small, find the probability that it does not have blemishes. [3]
Question
Find the probability that a randomly selected apple is classified as small. [2]
Hint
Hint
Hint
Use the normal cumulative distribution function with μ = 160 and σ = 15; calculate the z-score for 140 g.
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Complete Answer
Approximately 0.0912
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Question
Find the probability that a randomly selected apple is classified as medium. [2]
Hint
Hint
Hint
Find z-scores for 140 g and 175 g and compute the probability between these values.
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Approximately 0.750
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Question
Out of a sample of 200 randomly selected apples, find the expected number of apples classified as medium. [2]
Hint
Hint
Hint
Multiply the total number of apples (200) by the probability of an apple being medium (from part b).
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Approximately 150
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Question
Find the value of w such that 10% of apples are classified as large. [3]
Hint
Hint
Hint
Since 10% are large, P(W > w) = 0.10, so find w for which P(W ≤ w) = 0.90 using the inverse normal function.
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Approximately 179 g
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Question
Find the probability that at least 100 of these apples are classified as small. [3]
Hint
Hint
Hint
Let X ~ Bin(500, 0.0912) and use the binomial cumulative distribution function to find P(X ≥ 100).
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Approximately 1.55×10⁻¹¹
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Question
Find the probability that the selected apple is small and has blemishes, assuming weight and blemishes are independent events. [2]
Hint
Hint
Hint
Multiply the probability of the apple being small by the probability of having blemishes (0.05).
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Approximately 0.00456
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Question
Given that a selected apple is small, find the probability that it does not have blemishes. [3]
Hint
Hint
Hint
Use the independence of blemishes and weight; calculate P(no blemish | small) = 1 – 0.05.
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