Question 45·SAT Desmos Regression Practice·Problem Solving and Data Analysis

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The scatterplot shows the weekly sunlight exposure and weekly growth of nine plants. The point labeled $R$ represents a plant that was grown in unusually poor soil. A line of best fit for all nine points is shown.

Suppose point $R$ were omitted and a new line of best fit were drawn using the remaining eight points. Compared with the line shown, which choice accurately describes the expected changes to the slope and the $y$ -intercept of the new line of best fit?

An outlier is a point far from the overall pattern, and removing one can change a line of best fit. When the remaining points follow a straight path, compare that path with the drawn line. Use Desmos to calculate each slope from points on its line, then compare their $y$ -intercepts. Don't use the outlier as a point on either line.

Hints

Hint 1
The slope measures how much growth rises for each extra hour of sunlight. Use two points on the drawn line, rather than two plant dots, to find its slope.
Hint 2
Without $R$ , the other plants follow one straight path. Pick two well-separated points on that path. How much does growth rise, and how much does sunlight increase between them?
Hint 3
The $y$ -intercept is where a line meets the vertical axis, at zero hours of sunlight. Once you know the new slope, use a point on the new line to find that intercept.

Step-by-step

Compare the drawn line with the eight-point trend
Step 1
Find the slope of the drawn line
The drawn line passes near $(0,23)$ and $(50,88)$ . Its slope is the rise in growth divided by the increase in sunlight. Type $\frac{88-23}{50-0}$ into Desmos; it shows about $1.3$ millimeters per hour. The line meets the vertical axis near $23$ , so its $y$ -intercept is about $23$ .
Step 2
Find the slope without $R$
The eight circular points lie on one straight path. Choose $(5,20)$ and $(40,90)$ , two well-separated points on it. Type $\frac{90-20}{40-5}$ into Desmos; it shows $2$ millimeters per hour for the new slope. Don't include $R$ : its unusually low growth puts it below this path.
Step 3
Find the new intercept
A line with slope $2$ has the form $y=2x+b$ , where $b$ is its $y$ -intercept. The new line passes through $(5,20)$ , so substitute that point:
$20=2(5)+b.$
Subtract $2(5)$ from both sides:
$b=20-2(5).$
Type $20-2(5)$ into Desmos; it shows $10$ .
Step 4
Compare both parts of the lines
Compare the slope and intercept separately. The new slope, $2$ , is greater than the drawn line's approximately $1.3$ . The new intercept, $10$ , is less than its approximately $23$ . Removing the poor-soil plant makes the fitted line steeper and lowers where it meets the growth axis. Choice B.

Question 45·SAT Desmos Regression Practice·Problem Solving and Data Analysis

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