Question 869·Hard·Math

0 / 2000

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?

A residual compares a measured point with the model's prediction at the same depth: observed intensity minus predicted intensity. If the line of best fit is already given, use that equation rather than fitting a new line. Enter the plotted points and the equation in Desmos to calculate the differences. Greatest means the largest signed difference, not the largest distance from the line.

Hints

Hint 1
A data point gives two readings: its horizontal coordinate is depth, and its vertical coordinate is observed intensity. Read the labeled dots into paired columns, keeping each depth beside its own intensity so you can match each result to a point.
Hint 2
The line of best fit predicts intensity at each depth. Give the supplied equation a function name in Desmos, then evaluate it at the four depths. Don't use the dots' heights as predictions; those are the observed values.
Hint 3
A residual is observed intensity minus predicted intensity. Its sign matters: a point below the line has a negative residual even if it's far from the line. Which computed difference is greatest without taking absolute values?

Step-by-step

Compare the residuals in Desmos
Step 1
Pair each depth with its observed intensity
Read the labeled dots as $A=(8,25)$ , $B=(12,20)$ , $C=(15,8)$ , and $D=(18,10)$ . The vertical coordinate is the observed intensity. Enter the depths in Desmos's $x_1$ column and the intensities in its $y_1$ column. Keep the rows in this order so each calculated difference stays matched to its letter.
Step 2
Store the supplied line as a function
The line gives a prediction, the intensity the model expects at a depth. Type $g(x)=-2x+42$ to store the given line as $g$ . Desmos can now use $g$ for every table depth; fitting another line would change the predictions the question asks you to use.
Step 3
Find the prediction at each depth
Type $g(x_1)$ . Desmos returns $[26,18,12,6]$ , one predicted intensity for each row from A through D. These come from the line, not from the measured dots.
Step 4
Subtract and compare signed residuals
The given residual rule says observed minus predicted. Type $d_1=y_1-g(x_1)$ ; Desmos returns $[-1,2,-4,4]$ for A through D. The greatest residual is the largest signed value, not the greatest distance from the line. Point D's residual is $4$ intensity units, the greatest of the four. Choice D.

Question 869·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this