Question 2
The table shows the time a technician has spent servicing photocopiers.
| Time (in minutes) | Frequency |
|---|---|
| $0\le t\lt5$ | 10 |
| $5\le t\lt10$ | 20 |
| $10\le t\lt15$ | 30 |
| $15\le t\lt20$ | 40 |
| $20\le t\lt25$ | 100 |
What is the probability that a given service required at least 10 minutes but less than 20 minutes?
Not quite. 0.15 is $\dfrac{30}{200}$, the $10\le t\lt15$ row on its own. “Less than 20 minutes” also takes in the $15\le t\lt20$ row, so add its 40 as well. In 2021, 3.1% of Queensland students chose (A).
That’s it. Step through the hints below to see why. In 2021, 67.9% of Queensland students chose (B).
Not quite. 0.70 comes from two different slips. If you added the right rows, $30+40=70$, then divided by 100, remember there are 200 services in the table, so divide by the total of all five frequencies. If you got $\dfrac{140}{200}$, you used the $15\le t\lt20$ and $20\le t\lt25$ rows. The question starts at 10 minutes and stops before 20. In 2021, 27.2% of Queensland students chose (C).
Not quite. 0.85 is $\dfrac{170}{200}$, every service that took at least 10 minutes. That includes the $20\le t\lt25$ row, but the question stops at less than 20 minutes. In 2021, 1.5% of Queensland students chose (D).
This is the interval in the question. “At least 10” includes 10, and “less than 20” stops just before 20, so it covers exactly the $10\le t\lt15$ and $15\le t\lt20$ rows: 70 of the 200 services.
This interval gives , the value in option . It isn’t the interval the question describes, which runs from 10 up to but not including 20 minutes.
Every service is in this interval, so the probability is 1. That’s why the total of all the frequencies, 200, is what you divide by.
This interval holds of the 200 services. Move the sliders to the interval in the question.
A probability from a frequency table is a relative frequency: the frequency you want, divided by the total frequency. Start with the total.
$10+20+30+40+100$$=200$
“At least 10 minutes” means $t\ge10$. “Less than 20 minutes” means $t\lt20$. Together they give $10\le t\lt20$, which is exactly two rows of the table.
$10\le t\lt15:\ 30$$15\le t\lt20:\ 40$
Add the two frequencies and divide by the total.
$$\frac{30+40}{200}=\frac{70}{200}=0.35$$
Sense check: half the services, 100 of 200, are in the $20\le t\lt25$ row. So the chance of anything under 20 minutes is at most 0.5, which rules out 0.70 and 0.85.
The total frequency is $10+20+30+40+100=200$. “At least 10 but less than 20 minutes” is the rows $10\le t\lt15$ and $15\le t\lt20$.
$P(10\le t\lt20)$$=\dfrac{30+40}{200}=0.35$
How Queensland students answered in 2021
From the QCAA Mathematical Methods subject report for 2021. 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 2021 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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