Question 10
Handspans of teenagers are approximately normally distributed, with a mean of 15 cm and a standard deviation of 2 cm.
Which of the following groups is expected to be the largest?
Not quite. 7 cm to 11 cm is 2 to 4 standard deviations below the mean, out in the tail. Fewer than 2.5% of teenagers are there. In 2021, 1.9% of Queensland students chose (A).
Not quite. 11 cm to 15 cm runs from 2 standard deviations below the mean up to the mean, about 47.5%. Group (C) is the same width but centred on the mean, where the curve is tallest, so it holds about 68%. In 2021, 3.5% of Queensland students chose (B).
That’s it. Step through the hints below to see why. In 2021, 83.1% of Queensland students chose (C).
Not quite. 17 cm to 21 cm is 1 to 3 standard deviations above the mean. That’s about $13.5\%+2.35\%\approx16\%$. In 2021, 11.0% of Queensland students chose (D).
Centred on the mean, this window holds about 68%. No other 4 cm window holds more, because it covers the tallest part of the curve. That’s group (C).
This is group (B), from 2 standard deviations below the mean up to the mean, about 47.5%. It’s as wide as (C), but it sits to one side of the peak.
This is group (D), 1 to 3 standard deviations above the mean, about 16%.
This is group (A), 2 to 4 standard deviations below the mean. It’s out in the tail, with fewer than 2.5% of teenagers.
Very close, but not centred. A 4 cm window holds the most when its middle is exactly 15 cm. Use the button below.
This window holds about of teenagers. Its middle is cm, so slide it , toward the mean, to catch more.
It holds the same proportion as group (), its mirror image about the mean.
Turn each boundary into a $z$-score with $z=\dfrac{x-15}{2}$.
| (A) | $z=-4$ to $-2$ |
|---|---|
| (B) | $z=-2$ to $0$ |
| (C) | $z=-1$ to $1$ |
| (D) | $z=1$ to $3$ |
Every group is 4 cm wide, which is 2 standard deviations.
Use the empirical rule. About 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. Halve each band by symmetry.
(C) is exactly the middle band, about 68%. (B) is half of the 95% band, about 47.5%.
(D) is $13.5\%+2.35\%\approx16\%$, and (A) is less than the 2.5% beyond 2 standard deviations below the mean.
So (C) is the largest. Same width, but centred on the mean, where the curve is tallest.
$z=\dfrac{13-15}{2}=-1$$\text{and}\ z=\dfrac{17-15}{2}=1$
About 68% of teenagers are within 1 standard deviation of the mean. The other groups hold about 47.5% (B), 16% (D) and less than 2.5% (A).
How Queensland students answered in 2021
From the QCAA Mathematical Methods subject report for 2021. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2021 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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