Question 759·Hard·Math

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Each dot in the scatterplot above represents an apartment listing in a city, where $x$ is the distance from downtown (in kilometers) and $y$ is the monthly rent (in hundreds of dollars). A line of best fit for the data is shown, and one data point is labeled $P$ .

The residual for a point is defined as

(\text{actual rent})-(\text{rent predicted by the line of best fit}).

Suppose the monthly rent for the apartment represented by point $P$ decreases by $200, and the line of best fit is shifted upward by $0.4$ (hundreds of dollars) at every $x$ -value.

Which choice gives the new residual for point $P$ , in dollars?

A residual is a dot’s actual $y$ -value minus the fitted line’s predicted $y$ -value at the same $x$ . If the scatterplot already shows a line of best fit, use points on that line to write its rule and evaluate it in Desmos; don’t fit a new line through the dots. A lower actual value and a higher prediction both lower the residual. Check the axis units before giving the answer.

Hints

Hint 1
A dot gives the actual rent, while the drawn line gives the predicted rent. The open circles are on the line, so which two coordinates can you use to find its prediction without fitting the data again?
Hint 2
The slope is how much a line’s $y$ -value changes for each $1$ -unit increase in $x$ . Use the two points on the drawn line to write its rule, then evaluate it at point $P$ ’s distance.
Hint 3
The residual is actual minus predicted. Convert the rent decrease to hundreds of dollars before subtracting it from the dot’s rent, and add the upward shift to the line’s prediction. What unit must your answer use?

Step-by-step

Read the line, then update the residual
Step 1
Separate the dot from the line
At $x=8$ , the dot labeled $P$ has $y=15.5$ , so its actual rent is $15.5$ hundreds of dollars. The open circles at $(0,20)$ and $(12,11)$ lie on the drawn line, not among the apartment listings. Use those circles to find the line’s prediction.
Step 2
Write the line’s prediction rule
Call the predicted rent $f(x)$ . The line starts at $20$ when $x=0$ and changes by $11-20$ over $12$ kilometers, so its slope is $\frac{11-20}{12}$ . Type $f(x)=20+\frac{11-20}{12}x$ in Desmos. Desmos stores the rule for the drawn line; you don’t need to refit the dots.
Step 3
Find the prediction at point P
Type $f(8)$ . Desmos shows $14$ , so the line predicts $14$ hundreds of dollars at $P$ ’s distance. Keep this separate from the dot’s actual value, $15.5$ .
Step 4
Set up the changed residual
The vertical axis counts hundreds of dollars, so a $200 decrease is $2$ on that scale. The new actual value is $15.5-2$ , and the shifted line predicts $14+0.4$ . Using the given definition, actual minus predicted, write:
$r_{\text{new}}=(15.5-2)-(14+0.4)\quad\text{hundreds of dollars}.$
Lowering the actual value and raising the prediction both lower the residual.
Step 5
Calculate in the requested unit
Type $100((15.5-2)-(f(8)+0.4))$ to calculate the residual and convert hundreds of dollars to dollars. Desmos shows $-90$ . The apartment’s new rent is $90 below the shifted line’s prediction, so its new residual is $-90$ dollars. Choice A.

Question 759·Hard·Math

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