Question 833·Hard·Math

0 / 2000

The scatterplot above shows the relationship between practice time and quiz score for 9 students. A line of best fit is drawn and is labeled with the equation $y=1.0x+62.3$ .

The point labeled P was recorded incorrectly and should be removed. If P is removed and a new line of best fit is drawn for the remaining points, which choice is most likely the equation of the new line of best fit?

A line of best fit follows the overall trend of a scatterplot, not every dot. When a recorded point must be removed, fit only the remaining observations. Read their coordinates, then use a Desmos linear regression to find the new slope and intercept. A low point far to the right can flatten the original line, so don't reuse the old equation.

Hints

Hint 1
Each dot pairs one student's practice hours with that student's quiz score. The horizontal coordinate gives hours, and the vertical coordinate gives points. Since P was recorded incorrectly, leave its pair out when you enter the data.
Hint 2
A linear regression finds a line that follows several points closely, even when it doesn't pass through every dot. Fit the remaining pairs with $y_1\sim mx_1+b$ . Then compare both numbers Desmos reports with the equations offered.

Step-by-step

Approach 1: Fit the remaining points in Desmos

Step 1
Enter only the valid pairs
Read each dot as $(\text{practice hours},\text{quiz score})$ . Enter the eight pairs from $x=1$ through $x=8$ in a Desmos table, keeping each student's hours and score in the same row. Omit P at $(9,50)$ : the question says that recording is wrong, so it cannot count toward the new line.
Step 2
Find the new slope and intercept
Type $y_1\sim mx_1+b$ below the table. The $\sim$ tells Desmos to fit a line to the pairs. Under PARAMETERS, it reports $m\approx3.31$ and $b\approx54.61$ . Here $m$ is the predicted score increase per extra hour, and $b$ is the predicted score at zero hours.
Step 3
Match the whole equation
The new fit is approximately $y=3.31x+54.61$ . Compare both the slope and the intercept: the closest offered equation is $y=3.2x+54$ . The original slope of $1.0$ came from a fit that still included P, so it is too flat for the remaining students. The new line predicts about $3.2$ more quiz points per practice hour. Choice A.

Approach 2: Estimate from two distant dots

Step 1
Estimate the rise per hour
For a quick estimate, use two far-apart dots on the remaining upward pattern: $(1,58)$ and $(8,80)$ . The slope is the score change divided by the hours change. Type $m=(80-58)/(8-1)$ ; Desmos shows $m\approx3.14$ points per hour.
Step 2
Estimate the starting score
In $y=mx+b$ , the intercept $b$ is the predicted score at zero hours. Use the dot $(1,58)$ with the estimated slope: $58=m(1)+b$ .Subtract $m(1)$ : $b=58-m(1)$ .Type $b=58-m(1)$ below the earlier line. Desmos shows $b\approx54.86$ .
Step 3
Choose the closest model
Two dots give an estimate, not an exact best-fit line for all eight dots. They suggest a slope near $3.14$ and an intercept near $54.86$ , so the closest equation for quiz score versus practice hours is $y=3.2x+54$ . Choice A.

Question 833·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this