2024 Paper 2, Q19
The normal distribution probability density function is
$$p(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}$$
with the parameters mean, $\mu$, and standard deviation, $\sigma$.
The speeds of electric scooter (e-scooter) riders on a particular section of a bike path are approximately normally distributed with a mean of 18 km/h. It is known that $p(10)=0.0135$.
The speed limit for e-scooters on this section of bike path is 23 km/h. A speed camera is set up and records the speeds of 75 e-scooter riders. Every rider travelling faster than the speed limit is given a $\$143$ fine. Before setting up the speed camera, the following suggestion was made.
The total of the fines expected to be issued will be more than $\$1500$.
Evaluate the reasonableness of this suggestion.
Seventy-five riders go past one speed camera
Most riders sit near 18 km/h. Only the right-hand tail gets fined, and a tail that thin cannot pay for the claim.
The density formula is there to be solved, not just substituted into
$p(10)=0.0135$ with $\mu=18$ leaves $\sigma$ as the only unknown. Substitute and solve on the GDC. There is nothing to differentiate or integrate.
$\sigma=4.0002$ and $\sigma=28.4020$ both satisfy it. One of them has to be rejected on physical grounds. And that rejection is worth a mark.
$P(X>23)$, times 75 riders, times $\$143$. The unfamiliar part is over as soon as $\sigma$ is known.
Hunt for the σ that makes p(10) = 0.0135
Drag $\sigma$. The height of the curve at $x=10$ rises, peaks and falls again. Which is exactly why the equation has two roots. Both are marked on the right.
Why is $\sigma=28.4$ rejected?
QCAA marking guide · 6 marks
$$0.0135=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac12\left(\frac{10-18}{\sigma}\right)^2}$$
Marker: correctly substitutes the known information, $x=10$ and $p(10)=0.0135$, into the given normal distribution formula.
Using a GDC, you get two solutions, $\sigma=4.0002$ and $\sigma=28.4020$.
Reject 28.4020: three standard deviations below the mean would be a negative speed, and three above would be over 100 km/h on an e-scooter.
Marker: determines a possible value for the standard deviation. FT marks allowed for errors in prior working.
$$X\sim N(18,4.00^2):\quad P(X\ge 23)=0.10566$$
Marker: determines the proportion of riders above 23 km/h. Appropriate rounding accepted, e.g. 0.106 or 0.11.
$$75\times 0.10566=7.92\ \text{riders}$$
Marker: determines the number of riders above 23 km/h. The decimal is accepted, and so is rounding to 7 or 8, and the later marks then follow from whichever you chose.
$$7.9245\times 143=\$1133.20$$
Marker: determines the expected total fines. Equivalent totals accepted for a rounded rider count, e.g. $8\times\$143=\$1144$ or $7\times\$143=\$1001$.
About $1133, which is less than $1500,,,, so the suggestion is not reasonable
Marker: provides an appropriate statement of reasonableness. This mark can only be awarded if previous evidence supports the conclusion. The sentence has to name the two numbers it compares.
The rejected root would have changed the verdict
$1133 · not reasonable
10.5 riders would have to be fined to reach $1500. The tail only supplies 7.9.
$4600 · reasonable
Keeping the wrong root reverses the conclusion, which is why the rejection is a marked step and not a formality.
At $\sigma=4$, $x=10$ is exactly two standard deviations out, so $p(10)=\frac{e^{-2}}{4\sqrt{2\pi}}=0.013499$. The paper chose 0.0135 to make the root land on a whole number.
The QCAA print the density function at the top of the question, and that is what makes it tricky. You are used to areas under the normal curve, not the height of it, so it is natural to ignore the formula and look for a probability instead. Here the height is the only way to find $\sigma$. This is a provided tool, because the formula is not something the syllabus asks you to know. Your calculator then gives you two roots, and there is a mark for choosing the right one. A standard deviation of 28 km/h around a mean of 18 km/h would put a quarter of the riders at a negative speed, which makes no sense. The rest is familiar, but the marker expects every step. Find the proportion, then the number of riders, then the money, then write your sentence. I see a lot of marks lost at that final sentence.
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.