Mathematical Methods — Question 15

QCAA 2024, Paper 2 · 4 marks

Q15 · 2024 · Technology-activeComplex familiar4 marks

QUESTION 15 (4 marks)

The term extremely tall is used to describe any person whose height is three standard deviations or more above the mean height of the population. A person who just qualifies as extremely tall in a country where heights are normally distributed with a mean height of 180 cm and a standard deviation of 10 cm travels to another country. The person discovers they are taller than exactly 90% of the destination country’s population. Assuming that the standard deviation of both countries is the same, determine the minimum height required to be considered extremely tall in the destination country.

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WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

Approximately

227.18

cm.

Worked solution

In the original country, the cut-off for extremely tall is

180+3(10)=210\text{ cm}.

For the destination country, the

z

-score for 90% area is

z=1.28155.

Thus

1.28155=\frac{210-\mu}{10},

giving

\mu=197.1845\text{ cm}.

The destination cut-off for extremely tall is therefore

\mu+3\sigma=197.1845+30=227.1845\approx227.18\text{ cm}.

correctly determines the cut-off for extremely tall in the original country

[1 mark]

correctly determines the z-score for a 90% area

[1 mark]

determines the mean height of the destination country

[1 mark]

determines the minimum height required in the destination country

[1 mark]

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Mathematical Methods — Question 15

QUESTION 15 (4 marks)

QCAA guide · typeset solution

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