Q19 · 2024 · Technology-activeComplex unfamiliar6 marks
QUESTION 19 (6 marks)
The normal distribution probability density function is with the parameters mean, , and standard deviation, .
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 .
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.
WORKED SOLUTION
6 marksQCAA guide · typeset solution
ANSWER
The suggestion is not reasonable. The expected total fines are approximately $1133.20.
Worked solution
Substitute and into the normal density function and solve for . This gives two mathematical solutions, approximately The value is not plausible for the stated rider-speed context, so use . Then The expected number of riders exceeding the limit is Therefore the expected total fines are Since this is less than $1500, the suggestion is not reasonable.
correctly substitutes the given information into the normal probability density function
determines possible standard deviation value(s)
selects the contextually appropriate standard deviation
determines the probability of exceeding the speed limit
determines the expected total fine value
provides an appropriate statement about the reasonableness of the suggestion
One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.
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