Question 11
The equation of a fitted line for the number of free throws in basketball, $t$, and the number of hours in a training session, $h$, is $t=26.781+12.974\,h$
The predicted number of free throws for a 5-hour training session, when rounded to the nearest whole number, is
Not quite. One way to get 64 is to make two slips at once: leave out the 26.781, which gives $12.974\times5=64.87$, then chop off the decimals. The intercept belongs in every prediction, and 64.87 would round up anyway. In 2022, 2.7% of Queensland students chose (A).
Not quite. 65 is $12.974\times5=64.87$ rounded, so one way to get it is to leave out the intercept. The fitted line starts at 26.781, and that number is part of the prediction for every value of $h$. In 2022, 6.1% of Queensland students chose (B).
Not quite. Your substitution is fine if you got 91.651, but that rounds up. One way to get 91 is to cut the number off at the decimal point. The first decimal digit is 6, so the nearest whole number is 92. In 2022, 9.6% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2022, 80.6% of Queensland students chose (D).
The fitted line is orange. Each option comes from two choices: keep the intercept or drop it, then round or chop. Try all four.
This is option (D). The intercept is in, and 91.651 rounds up because the first decimal digit is 6.
This is option (C). Chopping throws away the 0.651 without looking at it. Rounding to the nearest whole number looks at the 6 and goes up.
This is option (B). Without 26.781 you are on the blue dashed line, which starts at zero. That isn’t the line the question gave you, and every prediction comes out 26.781 too low.
This is option (A), both slips at once: the intercept is gone and the decimals of 64.87 are thrown away.
At of training the line predicts free throws. Move the slider back to 5 hours to match the question.
Match each letter to its meaning before you touch the calculator. $h$ is the number of hours in the session, so a 5-hour session means $h=5$. $t$ is what the line predicts: free throws.
The line has two parts. The 26.781 is the intercept, where the line starts. The 12.974 is the gradient, the extra free throws for each extra hour. Only the gradient gets multiplied by $h$, and the intercept stays in every prediction.
Substitute $h=5$ and type the whole thing in one line. The calculator does the multiplication before the addition.
$$\begin{aligned}t&=26.781+12.974\times5\\&=26.781+64.87\\&=91.651\end{aligned}$$
The question asks for the nearest whole number. 91.651 sits between 91 and 92, and the first decimal digit is 6, so it rounds up.
$$91.651\approx92\text{ free throws}$$
Check the size: about 27, plus about $13\times5=65$, is about 92. An answer in the 60s has lost the intercept.
$$\begin{aligned}t&=26.781+12.974\times5\\&=91.651\\&\approx92\text{ free throws}\end{aligned}$$
Keep the intercept, substitute $h=5$, and round to the nearest whole number at the end.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2022 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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