IB AA SL and HL: core and optional content Core: part(iv), both individual heights exceed1.5m. Optio

Question

IB AA SL and HL: core and optional content

Core: part(iv), both individual heights exceed1.5m. Optional: parts(i)–(iii),(v), and the comparison involving a total-height distribution.

The heights, in metres, of female and male students in a school follow normal distributions with the following data.

	Mean Height (metres)	Variance Height (metres${^2}$)
Female	$1.6$	$0.04$
Male	$1.75$	$0.05$

Find the probability that

Two randomly selected male are taller than twice a randomly selected female.
A randomly selected male student is taller than a randomly selected female student by at least $20$ cm.
The difference in height between a randomly selected male and a female is more than $30$ cm.
The heights of two randomly selected female students both exceed $1.5$ m.
The total height of two randomly selected female students is more than $3$ m.

Comment on the difference in answers for part (iv) and (v).

Modelling assumption for this exercise: heights of distinct selected students are mutually independent and follow the stated distribution for their group.

Accepted Answer

(i) $0.722$ (ii) $0.434$ (iii) $0.375$ (iv) $0.478$ (v) $0.760$ Comment: The event in (iv) is a proper subset of the event in (v), so (v) has the greater probability.

	Mean Height (metres)	Variance Height (metres\({^2}\))
Female	\(1.6\)	\(0.04\)
Male	\(1.75\)	\(0.05\)

Probability Problems Involving Normal Distributions of Student Heights

Probability Problems Involving Normal Distributions of Student Heights

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