Question 2·SAT Desmos Regression Practice·Problem Solving and Data Analysis

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The scatterplot shows the relationship between the number of minutes a 3D printer runs, $x$ , and the number of small parts it produces, $y$ , for several test runs.

Which choice is the best estimate of the slope of a line of best fit for the data?

A line of best fit follows the overall pattern of scattered measurements. To estimate its slope, enter the plotted points in Desmos and fit a line; the slope tells you the predicted change in parts per minute. Use the whole pattern, not one short stretch between nearby dots. That short stretch may be steeper than the line that fits all the data.

Hints

Hint 1
Each dot is an ordered pair: its horizontal coordinate gives minutes, and its vertical coordinate gives parts. Put minutes in Desmos’s $x_1$ column and parts in its $y_1$ column, keeping each pair in one row.
Hint 2
A linear regression fits one straight line to all the dots. Type $y_1\sim mx_1+b$ ; the tilde tells Desmos to fit the line. Which reported value is the slope?

Step-by-step

Approach 1: Fit the points in Desmos
Step 1
Enter the plotted pairs
Read the five dots as $(0,2)$ , $(2,6)$ , $(4,8)$ , $(6,11)$ , and $(8,14)$ . Enter them in a Desmos table, with minutes in $x_1$ and parts in $y_1$ . Keep each test run in one row so Desmos fits the right pairs.
Step 2
Find the fitted slope
Type $y_1\sim mx_1+b$ below the table. The tilde asks Desmos for a regression, a line fitted to all five dots. Under PARAMETERS, Desmos reports $m=1.45$ . Here $m$ is the slope, or predicted additional parts per minute.
Step 3
Match the estimate
Type $\frac{3}{2}$ on the next line. Desmos shows $1.5$ , which is closest to the fitted slope of $1.45$ . The estimated rate is about $\frac{3}{2}$ additional parts per minute. Choice B.

Approach 2: Estimate from the full span
Step 1
Choose dots far apart
No Desmos needed for a quick estimate from the plot. The dots $(0,2)$ and $(8,14)$ span the plotted minutes, so they capture the overall rise better than the short stretch from $(0,2)$ to $(2,6)$ .
Step 2
Set up rise over run
A slope is the change in $y$ divided by the change in $x$ . Subtract the coordinates in the same order on top and bottom: $m=\frac{14-2}{8-0}$ .
Step 3
Find the changes
Subtract the parts and minutes separately: $m=\frac{12}{8}$ . That’s a rise of $12$ parts over $8$ minutes, not $12$ parts per minute.
Step 4
Give the slope per minute
Reduce the fraction: $m=\frac{3}{2}$ . So the overall trend produces about $\frac{3}{2}$ additional parts per minute. Choice B.

Question 2·SAT Desmos Regression Practice·Problem Solving and Data Analysis

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