Question 15
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.
[4 marks]One person, two countries
The first country turns the definition into a height. The second country turns that height and a percentage into a mean. Then the definition runs again.
The person’s 210 cm links the two countries. In the new country they are 1.28 standard deviations above the mean, so they are 1.72 standard deviations short of extremely tall.
How far is the person from the new cut-off?
The person stays at 210 cm. The percentage of people they are taller than gives $z$, the number of standard deviations they sit above the mean. Extremely tall needs 3, so the cut-off is $(3-z)\sigma$ above them. Change the percentage and the standard deviation to see what each one controls.
This is the question. Taller than exactly 90% gives $z\approx1.2816$, so the mean is $210-12.8155\approx197.18$ and the person is $(3-1.2816)\times10\approx17.18$ cm short of the cut-off. The new minimum is $227.18$ cm.
Still 90%, so $z$ has not moved, but with a different standard deviation the mean and the cut-off both shift. The question tells you the standard deviations match. That assumption is what lets you use 10 cm twice, and it is what fixes the answer at 227.18 cm.
Taller than about 99.87% means $z=3$. The person sits exactly on the cut-off, so the gap $(3-z)\sigma$ is 0 whatever the standard deviation. With $\sigma=10$ this is the home country again.
Work through the solution one mark at a time
Try each step yourself before you reveal it. There is one calculator step, the inverse normal. Keep the unrounded values in your calculator until the last line.
Start in the original country. The definition puts the cut-off three standard deviations above the mean, and “just qualifies” means the person is exactly that tall.
$$\begin{aligned}\mu+3\sigma&=180+3\times10\\&=180+30\\&=210\ \text{cm}\end{aligned}$$
Marker: correctly determines the cut-off for extremely tall in the original country
Now the destination country. The person is taller than exactly 90%, so the area to the left of 210 cm is 0.90. The mean is the unknown, so sketch it as a question mark.
Work in the standard normal, where the mean is 0 and the standard deviation is 1. Use your calculator’s inverse normal function with an area of 0.9 to the left.
$$z\approx1.28155$$
This mark can be implied by your later working. Appropriate rounding is accepted, such as 1.28.
Marker: correctly determines the z-score for a 90% area
The person’s height is the $X$ value. The standard deviation is the same as at home, $\sigma=10$, so the mean is the only unknown in the $z$-score formula.
$$\begin{aligned}z&=\frac{X-\mu}{\sigma}\\1.28155&=\frac{210-\mu}{10}\\12.8155&=210-\mu\\\mu&=210-12.8155\\&=197.1845\ \text{cm}\end{aligned}$$
Appropriate rounding is accepted, such as 197.2. If your $z$-score had a slip, this mark can still be earned by follow-through.
Marker: determines the mean height of the destination country
Use the definition again, this time with the destination country’s mean.
$$\begin{aligned}&\mu+3\sigma\\&=197.1845+3\times10\\&=227.1845\end{aligned}$$
Appropriate rounding is accepted, such as 227.2. As a check, the person is 1.28155 standard deviations up and needs 3, so the cut-off is $(3-1.28155)\times10\approx17.18$ cm above 210 cm.
Marker: determines the minimum height required in the destination country
Each phrase of the question gave you something
Each piece is a standard normal-distribution move. What makes it complex familiar is that one person links two populations. You use the definition forwards in the first country to get a height, carry that height into the second country as the known value, run the unknown-mean routine from a percentage, and then use the definition forwards again with the new mean. The question never says “find the mean of the destination country”, so you have to see that it is the missing link. It stays familiar because the definition, the percentage and the matching standard deviation are all stated for you.
If I had to guess where you might lose marks, it would be the direction. Taller than 90% puts 210 cm above the new mean, so you subtract: $\mu=210-12.8155$. Adding gives a population where only 10% are shorter than the person. The next risk is stopping at 197.18, which earns three marks but answers a different question. The last mark needs the new cut-off, 30 cm above that mean.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording and marking-guide steps are from the 2024 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2024, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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