Question 5
Which normal distribution curve best represents a normal distribution with a mean of 1 and a standard deviation of 0.5?
Not quite. This curve has the right spread, but its peak is at $x=-1$. A normal curve peaks at its mean, so the peak must be at $x=1$. In 2022, 0.6% of Queensland students chose (A).
Not quite. Its peak is at $x=0.5$, but 0.5 is the standard deviation, not the mean. It’s also too wide: it spreads from about $-2.5$ to $3.5$, which fits a standard deviation of 1. In 2022, 2.0% of Queensland students chose (B).
Close. The peak is at the mean, $x=1$, but the curve is too wide. It spreads from about $-2$ to $4$, which fits a standard deviation of 1. With $\sigma=0.5$, almost all the area is between $1-1.5=-0.5$ and $1+1.5=2.5$. In 2022, 18.1% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2022, 78.9% of Queensland students chose (D).
A mean of 1 and a standard deviation of 0.5. Your curve matches (D): the peak is at $x=1$, and almost all the area is between $-0.5$ and $2.5$.
Your curve matches (A). The spread is right, but the mean is $-1$, not 1.
Your curve matches (B): a mean of 0.5 and a standard deviation of 1. Both are wrong for this question.
Your curve matches (C): the right mean, but a standard deviation of 1. It is twice as wide and half as tall as the curve you need.
The peak sits above the mean. A larger $\sigma$ makes the curve wider and lower, because the total area under it stays 1.
A normal curve is symmetric about its mean, so its peak sits directly above the mean. Here the peak must be at $x=1$.
That rules out (A), which peaks at $-1$, and (B), which peaks at $0.5$.
About 99.7% of a normal distribution lies within three standard deviations of the mean.
$1-3(0.5)=-0.5$$1+3(0.5)=2.5$
So the curve should be almost flat on the axis outside $-0.5$ to $2.5$.
(C) spreads from about $-2$ to $4$. That is $1\pm3(1)$, a standard deviation of 1.
(D) spreads from about $-0.5$ to $2.5$. That is $1\pm3(0.5)$. Halving the standard deviation also makes the curve twice as tall, which is why (D) is taller than (C).
The peak is at the mean, $x=1$, so the answer is (C) or (D).
$\mu\pm3\sigma=1\pm3(0.5)$$=-0.5\text{ to }2.5$
Almost all of the curve lies between $-0.5$ and $2.5$. Only (D) does that. (C) is a standard deviation of 1.
How Queensland students answered in 2022
From the QCAA Mathematical Methods subject report for 2022. 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 2022 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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