Mathematical Methods · Past QCAA questions All simple familiar questions
2022 · Paper 2 · Simple familiar

Question 12

[4 marks] Technology-active
Topic guidePremiumUnit 4 Topic 3.2 · Finding a probabilityFinding a quantile
Game plansPremium5.1 Finding a probability5.2 Finding a quantile
The question

Suppose that the distance travelled by vehicles in a year can be modelled by a normal distribution. In 2021, vehicles travelled a mean of 13 700 km with a standard deviation of 3400 km.

a)

Determine the probability that a vehicle chosen at random travelled less than 12 000 km in 2021.

[2 marks] b)

Determine the value of $x$ where 60% of vehicles travelled less than $x$ km in 2021.

[2 marks]
Watch the situation first

Distance in, area out. Then area in, distance out

Both parts use the same bell curve. Part a) gives you a distance and asks for an area. Part b) gives you an area and asks for the distance.

Mean 13 700 km, standard deviation 3400 km Part a): less than 12 000 km is the area to the left Part b): now the area is known, 60% to the left of x Inverse normal turns that area back into a distance Distance (km) μ σ = 3400 12 000 0.3085 60% x μ = 13 700, σ = 3400 12 000 − 13 700 = −1700 z = −1700 ÷ 3400 = −0.5 P(X < 12 000) ≈ 0.3085 P(X < x) = 0.6 z ≈ 0.2533 14 561.38 x ≈ 14 561.38 km

12 000 km is exactly half a standard deviation below the mean, so less than half the vehicles fall below it. 60% is just over half, so $x$ sits just above the mean.

Now for the mathematics

Decide what you know before you reach for the calculator

Both parts turn on one decision: are you given the distance or the area? Choose what you know, then move the slider. The orange area is the proportion of vehicles that travelled less than the black line.

μ Distance (km)
What you know

This is part a). 12 000 km is exactly half a standard deviation below the mean, so the area to its left is less than a half: about 0.3085.

This distance is below the mean, so less than half the area is to its left.

At the mean, exactly half the area is to the left, so the probability is 0.5.

This distance is above the mean, so more than half the area is to its left.

This is part b). An area of 0.6 is just over a half, so $x$ sits just above the mean, at about 14 561.38 km.

An area under a half to the left puts the distance below the mean.

An area of exactly a half puts the distance at the mean, 13 700 km.

An area over a half to the left puts the distance above the mean.

Distance in, probability out. That is your calculator’s normal cumulative distribution function.

Probability in, distance out. That is your calculator’s inverse normal function, given the area to the left.

QCAA marking guide · 4 marks

Work through the solution one mark at a time

Try each step yourself before you reveal it. Each part has one mark for showing what you are finding and one mark for the number your calculator gives.

Step 1 · part a)1 mark

Let $X$ be the distance a vehicle travelled in 2021. “Less than 12 000 km” is the area under the normal curve to the left of 12 000. Sketch it, or write it as a probability statement.

μ = 13 700 12 000

$$P(X<12\,000)$$

The guide accepts a diagram or equivalent statements, such as finding the area to the left of 12 000 under a normal curve, or $P(x<12\,000)$.

Marker: correctly uses an appropriate mathematical representation

Step 2 · part a)1 mark

You know the distance and want the area, so use your calculator’s normal cumulative distribution function. It finds the area between a lower and an upper bound. “Less than” has no lower limit, so give a lower bound far below the mean, such as a very large negative number. The upper bound is 12 000, the mean 13 700 and the standard deviation 3400.

$P(X<12\,000)\approx0.3085$

Equivalent decimal values are accepted, such as 0.308538. As a check, $z=\frac{12\,000-13\,700}{3400}=-0.5$, so 12 000 is below the mean and the answer must be less than 0.5.

Marker: correctly determines the probability

Step 3 · part b)1 mark

This time the area is given. “60% of vehicles travelled less than $x$ km” means the area to the left of $x$ is 0.6, and $x$ is the unknown upper limit.

μ = 13 700 x 0.6

$$P(X<x)=0.6$$

The guide accepts a diagram or equivalent statements, such as finding the upper limit of the interval of area 0.60, or $P(x<a)=0.6$.

Marker: correctly uses an appropriate mathematical representation

Step 4 · part b)1 mark

You know the area and want the distance, so use your calculator’s inverse normal function. Give it the area to the left, 0.6, the mean 13 700 and the standard deviation 3400. If it asks which tail, choose the left, because the 60% is below $x$.

$x\approx14\,561.38$ km

Equivalent decimal values are accepted, such as 14 561.3802. You can also find $z\approx0.2533$ from the standard normal first, then $x=13\,700+0.2533\times3400$. The area is just over a half, so $x$ should be just above the mean, and it is.

Marker: correctly determines the value of $x$

Putting it all together

Each phrase of the question gave you something

“can be modelled by a normal distribution”
Probabilities are areas under a bell curve.
“a mean of 13 700 km with a standard deviation of 3400 km”
$\mu=13\,700$ and $\sigma=3400$, the same in both parts.
“travelled less than 12 000 km”
The area to the left of 12 000, $P(X<12\,000)\approx0.3085$.
“60% of vehicles travelled less than $x$ km”
The area to the left of $x$ is known, $P(X<x)=0.6$, so use inverse normal.
“Determine the value of $x$”
$x\approx14\,561.38$ km, just above the mean.
What makes this simple familiar

Each part is one calculator function. Part a) goes forwards, from a distance to an area. Part b) goes backwards, from an area to a distance. Nothing needs rearranging, and the mean and standard deviation are given.

If I had to guess where you might lose marks, it would be the representation. Half of each part’s marks are for showing what you are finding, so sketch the curve or write $P(X<12\,000)$ before you reach for the calculator. The other risk is the side of the curve. Taking the area to the right gives 0.6915, and an area of 0.4 gives about 12 838.62 km, which is below the mean. A quick check against the mean catches both.

Only mark this done when you could do it without help. Reading the solution does not count.

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Question wording and marking-guide steps are from the 2022 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.

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