Infer a normal standard deviation from a tail probability and use it for an expectation count.

Infer a normal standard deviation from a tail probability and use it for an expectation count.

9 marks
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The heights of trees in a forest are normally distributed with mean height 17 metres. One tree is selected at random. The probability that a selected tree has a height greater than 24 metres is 0.06.

  1. Find the probability that the tree selected has a height less than 24 metres.[2]
  2. The probability that the tree has a height less than D metres is 0.06. Find the value of D.[3]
  3. A woodcutter randomly selects 200 trees. Find the expected number of trees whose height lies between 17 metres and 24 metres.[4]

Solution:

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Answer:\(P(17<X<24)=0.5-0.06=0.44\), so the expected number among 200 trees is \(200(0.44)=88\).

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