A scientist measured light intensity, , at different depths, , in a lake. A scatterplot of the data is shown, with four labeled points , , , and . The line of best fit for the data is given by For any data point, the residual is defined as Which labeled point has the greatest residual?

Solution available with step-by-step explanation

SAT Two-Variable Data Question 51 | Free SAT Prep | Aniko

00:00

Question 51·Hard·Two-Variable Data: Models and Scatterplots

0 / 57

A scientist measured light intensity, $y$ , at different depths, $x$ , in a lake. A scatterplot of the data is shown, with four labeled points $A$ , $B$ , $C$ , and $D$ .

The line of best fit for the data is given by

y=-2x+42

For any data point, the residual is defined as

\text{residual}=\text{observed }y-\text{predicted }y.

Which labeled point has the greatest residual?

When a line of best fit is given, treat it as a prediction rule: plug each point’s $x$ into the line to get $y_{\text{predicted}}$ , then subtract to get the residual. To answer “greatest residual,” compare the residual values directly (most positive wins), not the absolute distances from the line.

Hints

Click me!

Compute predicted values from the line

Use the given line of best fit $y=-2x+42$ to compute a predicted $y$ for each labeled point’s $x$ -value.

Residual means observed minus predicted

Be careful with the order: residual $=y_{\text{observed}}-y_{\text{predicted}}$ . A point above the line will have a positive residual.

Compare all four residuals

After computing the four residuals, select the point with the largest numerical value (the most positive residual).

Desmos Guide

Enter the line and the points

In Desmos, enter the line y=-2x+42. Then enter the points A=(8,25), B=(12,20), C=(15,8), and D=(18,10).

Compute predicted values on the line

Compute predicted values with yA=-2*8+42, yB=-2*12+42, yC=-2*15+42, and yD=-2*18+42.

Compute residuals and compare

Compute residuals with rA=25-yA, rB=20-yB, rC=8-yC, and rD=10-yD. The point with the largest residual is the correct choice.

Step-by-step Explanation

Set up residuals using the given model

For each labeled point, compute the predicted value from the line of best fit $y=-2x+42$ , then compute

\text{residual}=y_{\text{observed}}-y_{\text{predicted}}.

Compute the residual for each labeled point

From the graph:

Point $A$ is $(8,25)$ . Predicted: $-2(8)+42=26$ . Residual: $25-26=-1$ .
Point $B$ is $(12,20)$ . Predicted: $-2(12)+42=18$ . Residual: $20-18=2$ .
Point $C$ is $(15,8)$ . Predicted: $-2(15)+42=12$ . Residual: $8-12=-4$ .
Point $D$ is $(18,10)$ . Predicted: $-2(18)+42=6$ . Residual: $10-6=4$ .

Identify the greatest residual

The residuals are $-1$ , $2$ , $-4$ , and $4$ . The greatest residual is $4$ , which corresponds to Point D.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation