Maths q. What is the minimum height....?

Question

Background info>>>
The true height of men is known to be 173cm with a standard deviation of 6.5cm.
The proportion of men in the study are taller than 168cm is 78%.

Q1. Builders want to make doors to fit at least 99% of men, what is the minimum height (to nearest cm) they will need to make their doors?

Any help much appreciated. :)

Answer 1

We will assume that the height of men is normally distributed. Then, we can write this height as

X = 173 + 6.5 * Z

where Z is a standard normally distributed variable.

Here we are looking for a height x such that P(X < x) >= 0.99, in other words, such that the length of a random person is smaller than x with probability at least 99%.

P(X < x)
= P(173 + 6.5*Z < x)
= P(Z < [x - 173]/6.5)
= Phi([x - 173]/6.5)
= 0.99

where Phi is the cumulative distribution function of the standard normal distribution. Hence:

(x - 173)/6.5 =Phi^{-1}(0.99) = 2.326

or in other words

x = 173 + 6.5*2.326 = 188.12 cm

Rounding up, we arrive at x = 189 cm. To conclude, with the assumptions listed, the builders must design doors of at least 189 cm such that at least 99% of all men can walk through without crouching.

Answer 2

According to probability tables (see source below), the relevant height is 2.33 standard deviations above the mean.

2.33 * 6.5cm = 15cm (rounded to the nearest cm)
173cm + 15cm = 188cm

The doors need to be at least 188cm in height.

Answer 3

Make it simple. Use just simply 179.5 cm. I think it would cover fast all heights.

Answer 4

6 fts.