Question 17
Model bridges were constructed for a competition. The models that could support the heaviest loads before collapsing were given awards.
The load results of the competition were normally distributed, with a mean of 1.36 kg and a standard deviation of 0.12 kg.
Three award categories were used: honours for the top 15% of load results; distinction for the next 15%; and commendation for the next 15%.
The model bridge constructed by Finley only just missed out on a commendation. Kirby’s model bridge only just qualified for honours. Determine the difference, to the nearest gram, between the loads supported by Finley and Kirby’s models.
[5 marks]Three awards stacked from the top
Each award band sits under the one before it. Watch where Kirby and Finley land.
Commendation starts where the top 45% begins, because the three 15% bands stack up. Kirby sits on the honours cut-off and Finley sits on the commendation cut-off.
Choose the area that puts Finley on the right line
Kirby’s cut-off is fixed by the top 15%. Choose the area you would give the inverse normal for Finley, and see which line it lands on. Then see what rounding the cut-offs does to the answer in grams.
Choose an area. Finley only just missed a commendation, so Finley’s load sits right on the lowest commendation load.
An area of 0.15 to the right is the honours cut-off. That is Kirby’s line, so the difference comes out as 0. Finley is two bands further down.
An area of 0.30 to the right is the lowest distinction load. Finley missed commendation, the band below distinction, so one more 15% has to stack on.
Honours, distinction and commendation together are the top 45%, so the lowest commendation load has 0.45 to its right. Finley sits on that line, and the difference, with unrounded cut-offs, is about 109 g.
This puts 45% of loads below Finley instead of above. The awards go to the heaviest loads, so the 0.45 belongs in the right tail. An area of 0.55 to the left gives the same line as 0.45 to the right.
With 1.48 and 1.38 the difference is 100 g, which is 9 g out. Those rounded cut-offs still earn their own marks, but the answer to the nearest gram needs the unrounded values.
Work through the solution one mark at a time
Try each step yourself before you reveal it. Your calculator’s inverse normal gives both cut-offs. A diagram helps you choose the areas, but it is not needed for the marks.
Let $X$ be the load in kilograms, so $X\sim N\left(1.36,0.12^2\right)$. Kirby is on the lowest honours load, the top 15%. Finley is on the lowest commendation load. The three awards stack, so that is the top 45%.
$$\begin{aligned}\text{Area 1}&=0.15\\\text{Area 2}&=3\times0.15\\&=0.45\end{aligned}$$
Both are right-tail areas. This mark may be implied by your later working, and the left-tail areas 0.85 and 0.55 are accepted too.
Marker: correctly identifies both required areas
Use your calculator’s inverse normal with $\mu=1.36$ and $\sigma=0.12$. Kirby only just qualified for honours.
$$\begin{aligned}P(X>x)&=0.15\\\text{or }P(X<x)&=0.85\\x&\approx1.48437\ \text{kg}\end{aligned}$$
An answer in grams is accepted, and so are equivalent decimal values such as 1.48.
Marker: correctly determines honours lowest load
Now the lowest commendation load, the top 45%. Finley only just missed it, so Finley’s load is this value.
$$\begin{aligned}P(X>x)&=0.45\\\text{or }P(X<x)&=0.55\\x&\approx1.37508\ \text{kg}\end{aligned}$$
An answer in grams is accepted, and so are equivalent decimal values such as 1.38.
Marker: correctly determines commended lowest load
Subtract, keeping the unrounded cut-offs.
$$\begin{aligned}&1.48437-1.37508\\&=0.10929\ \text{kg}\end{aligned}$$
If either cut-off had a slip, this mark can still be earned by follow-through.
Marker: determines the difference in loads
The question asks for the nearest gram. There are 1000 g in a kilogram.
$$0.10929\times1000=109.29\ \text{g}$$
Subtracting the rounded cut-offs 1.48 and 1.38 gives 100 g instead, so keep the unrounded values until here.
Marker: converts answer to the nearest gram
Each phrase of the question gave you something
Each cut-off on its own is a routine quantile. The work is in turning the story into areas. You have to stack the three 15% bands to see that commendation starts at the top 45%, and read “only just qualified” and “only just missed out” as sitting on a cut-off. Then two quantiles feed a subtraction and a change of units. It is still familiar because the question names the normal distribution, the mean, the standard deviation and every percentage you need.
If I had to guess where you might lose marks, it would be the area for Finley. The first mark is for both areas, and using 0.15 or 0.30 for the commendation band puts Finley on the wrong line. The other risk is rounding too early. The guide accepts 1.48 and 1.38 for the two cut-off marks, but subtracting those gives 100 g. Keep five or more decimal places until the last line, then convert to grams.
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 2023 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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