Middle School Math Quiz: Fit Lines To Scatter Plots

20 questions · exam conditions

0:00

Fit Lines To Scatter PlotsQuestion 1 of 20

A scatter plot shows the relationship between the number of laps swum ( $x$ ) and total time in minutes ( $y$ ). A student drew a line of best fit.

To judge how well the line fits, which distances should you mainly look at?

The horizontal distances from the points to the line (how far left or right each point is from the line).

The distance from each point to the origin.

The vertical distances from the points to the line (how far up or down each point is from the line).

Only whether the line touches at least two points.

← Back to quizzes

Middle School Math Quiz

Middle School Math Quiz: Fit Lines To Scatter Plots

Practice Fit Lines To Scatter Plots in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Fit Lines To Scatter Plots, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A scatter plot shows the relationship between the number of laps swum ( $x$ ) and total time in minutes ( $y$ ). A student drew a line of best fit.

To judge how well the line fits, which distances should you mainly look at?

The horizontal distances from the points to the line (how far left or right each point is from the line).
The distance from each point to the origin.
The vertical distances from the points to the line (how far up or down each point is from the line). (correct answer)
Only whether the line touches at least two points.

Explanation: Tests informally fitting straight line to scatter plot showing linear association and assessing fit by judging point closeness to line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of laps swum vs time, assessing fit by measuring vertical distances from points to the line, judging smaller averages as better. The correct distances to look at are the vertical distances from the points to the line (choice A). A common error is using horizontal distances or distances to the origin, but vertical residuals matter for predicting y from x. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 2

A student drew a line of best fit for a scatter plot of $x$ = practice minutes and $y$ = free-throw percentage. The student wants to check the fit.

Which method is the best way to judge whether the line fits well?

Check whether the line touches the highest and lowest points
Check whether the points are vertically close to the line (small up-and-down distances) (correct answer)
Check whether the points are horizontally close to the line (small left-and-right distances)
Check whether the line passes through every point

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of practice minutes vs free-throw percentage showing upward linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points within ±5% of line vertically (good fit), or comparing two possible lines where one passes closer to majority of points (better fit). The correct method is to check whether the points are vertically close to the line (small up-and-down distances), as this measures prediction accuracy for y-values. A common error is judging by horizontal distances, but fit is assessed vertically since the line predicts y from x. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 3

A scatter plot shows the relationship between number of pages read ( $x$ ) and minutes spent reading ( $y$ ). A student drew the line $y=2x+5$ as a model.

How good is the fit of the line $y=2x+5$ ?

Poor fit, because the line must pass through every point to be a good fit
Good fit, because you should judge fit using how close the points are horizontally to the line
Good fit, because most points are within about 1–2 minutes vertically from the line (correct answer)
Poor fit, because many points are about 10–20 minutes away from the line

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of pages read vs minutes spent reading showing upward linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points within ±2 minutes of line vertically (good fit), or comparing two possible lines where one passes closer to majority of points (better fit). The correct assessment is a good fit, because most points are within about 1–2 minutes vertically from the line, indicating accurate predictions. A common error is claiming poor fit because the line doesn't pass through every point, but lines approximate trends in scatter plots, allowing for some deviation. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 4

A class measured how many pages they read ( $x$ ) and how many minutes it took ( $y$ ). The scatter plot shows a clear positive linear trend.

A student drew the line $y=2x+5$ on the scatter plot. Most points are within about 3 minutes vertically of the line.

How would you describe the fit of this line?

Poor fit, because the line does not pass through every point.
Good fit, because the points are spread out widely from the line.
Poor fit, because the points are above the line more often than below it.
Good fit, because most points are close to the line (small vertical distances). (correct answer)

Explanation: Tests informally fitting straight line to scatter plot showing linear association and assessing fit by judging point closeness to line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of pages read vs minutes taken showing positive linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points within ±3 minutes of line vertically (good fit), or comparing to alternatives where points are farther (poorer fit). The correct assessment is good fit because most points are close to the line with small vertical distances (choice A). A common error is claiming poor fit because the line does not pass through every point, but scatter plots show variability and lines approximate the trend. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 5

A student drew a line of best fit for a scatter plot of (x = number of texts sent, y = phone battery percent used). One point is far away from the rest (an outlier). Which choice best describes how the line should be drawn?

Draw the line to fit the main cluster of points, without letting the one outlier determine the slope. (correct answer)
Draw a horizontal line, because outliers mean there is no relationship at all.
Draw a curved line so it hits every point exactly, including the outlier.
Draw the line through the outlier and ignore the rest of the points.

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of texts sent vs battery used with one outlier, fitting line through the main cluster's center, assessing fit by closeness of majority points, ignoring the outlier's pull. The best approach is to draw the line to fit the main cluster of points, without letting the one outlier determine the slope. Common errors include drawing through the outlier only or using a curved or horizontal line. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 6

A scatter plot shows a positive linear association between the number of books checked out (x) and the total minutes spent reading (y). A student drew a line that goes above almost every point (most points are below the line). What is the best way to improve the line of best fit?

Ignore most points and draw the line through only the two highest points.
Move the line downward so it goes through the middle of the points, with a more balanced number of points above and below. (correct answer)
Make the line pass through every single point, even if it changes direction.
Rotate the line so it slopes downward, because lines of best fit should always slope down.

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of books checked out vs minutes reading showing upward linear trend, if a line is above most points, adjust it downward through the center for balance, improving fit by reducing vertical distances. The best improvement is to move the line downward so it goes through the middle of the points, with a more balanced number of points above and below. Common errors include forcing the line through all points or ignoring most points for extremes. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 7

A student plotted the number of push-ups completed (x) and the time in seconds to finish (y). The points show a negative linear trend (more push-ups means more time). Two lines are suggested. Which line is the better fit?

Line B, because a best-fit line should make all points fall above the line.
Line A, because it goes through the middle of the point cloud and leaves about the same number of points above and below. (correct answer)
Line B, because it is as steep as possible and passes through an extreme point.
Line A is worse, because a best-fit line must pass through every point exactly.

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of push-ups vs time showing downward linear trend, fitting line through approximate center sloping downward, assessing fit by observing most points within ±5 seconds of line vertically (good fit), or comparing two possible lines where one passes closer to majority of points (better fit). The correct choice is Line A, because it goes through the middle of the point cloud and leaves about the same number of points above and below, capturing the negative trend. Common errors include prioritizing extreme points or expecting the line to pass through every point. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 8

A scatter plot shows the relationship between the number of text messages sent in a day ( $x$ ) and the number of minutes spent on the phone ( $y$ ). A student drew the line $y=-2x+120$ .

The points on the scatter plot clearly trend upward from left to right.

What is the best critique of the student's line?

The line is wrong because a best-fit line must go through every point.
The slope is the wrong direction; the line should have a positive slope to match the upward trend. (correct answer)
The line is correct because negative slopes always balance points above and below.
The line is wrong because it crosses the $y$ -axis.

Explanation: Tests informally fitting straight line to scatter plot showing linear association and assessing fit by judging point closeness to line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of text messages vs phone minutes with upward trend, a downward-sloping line fails to capture the positive association. The best critique is that the slope is the wrong direction; the line should have a positive slope to match the upward trend (choice A). A common error is accepting a negative slope because it balances points or crosses the y-axis, but the slope must match the trend direction. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 9

A scatter plot shows the relationship between distance biked (x) and time (y) for a student biking at a steady pace. Which statement best explains why a straight line is a reasonable model?

A straight line is not reasonable because distance and time are unrelated.
A straight line is reasonable because the line should pass through the highest and lowest points only.
A straight line is reasonable because the points show an upward linear trend. (correct answer)
A straight line is reasonable only if every point lies exactly on the line.

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of distance biked vs time showing upward linear trend at steady pace, fitting line through approximate center sloping upward, assessing fit by observing most points close to the line (good fit for linear relationship). The best explanation is that a straight line is reasonable because the points follow a roughly straight upward trend, showing a linear relationship. Common errors include thinking lines must hit every point or only connect extremes. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 10

A scatter plot shows the relationship between the number of items bought at a school store ( $x$ ) and the total cost in dollars ( $y$ ). A line is drawn on the plot.

If about 80% of the points are within about $\pm 1$ dollar vertically of the line, what is the best description of the line's fit?

Good fit, because most points are close to the line (small residuals). (correct answer)
Poor fit, because the line should be drawn through the highest and lowest points only.
Good fit, because points far from the line mean the line captures more variation.
Poor fit, because some points are not exactly on the line.

Explanation: Tests informally fitting straight line to scatter plot showing linear association and assessing fit by judging point closeness to line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of items bought vs cost showing linear trend, assessing fit as good if 80% of points are within ±1 dollar vertically of the line. The best description is good fit because most points are close to the line with small residuals (choice A). A common error is calling it poor fit because not all points are exactly on the line or suggesting the line should only connect extremes. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 11

A student drew a line of best fit for a scatter plot relating miles biked ( $x$ ) to minutes spent biking ( $y$ ). The line is $y=4x+2$ .

Use the line to estimate how many minutes it would take to bike 10 miles.

About 38 minutes
About 80 minutes
About 6 minutes
About 42 minutes (correct answer)

Explanation: Tests informally fitting straight line to scatter plot showing linear association and assessing fit by judging point closeness to line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of miles biked vs minutes spent showing positive linear trend, fitting line through approximate center, then using the line to predict y for a given x by plugging in the value. The correct prediction is about 42 minutes by substituting x=10 into y=4x+2 to get 42 (choice A). A common error is miscalculating the substitution, like ignoring the intercept or using the wrong operation. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 12

A scatter plot shows a linear trend, but there is one outlier point far away from the rest.

Which choice best describes how to draw an informal line of best fit?

Draw a curve so that it can pass through the outlier and the other points
Draw the line through the center of the main cluster of points, not letting the outlier control the line (correct answer)
Do not draw any line because one outlier means no relationship exists
Draw the line to go through the outlier exactly, even if it is far from most points

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot with a linear trend but one outlier, fitting line through approximate center of the main cluster sloping appropriately, assessing fit by observing most points within ±5 units of line vertically (good fit), ignoring the outlier's pull. The correct approach is to draw the line through the center of the main cluster of points, not letting the outlier control the line, to best represent the overall trend. A common error is drawing the line to go through the outlier exactly, but this distorts the fit for the majority of data. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 13

A scatter plot shows a positive linear trend between number of push-ups ( $x$ ) and points scored in a fitness game ( $y$ ).

Two students propose lines:

Line A: $y=3x+10$
Line B: $y=4x-5$

Data points: $(2,16), (3,19), (4,23), (5,26), (6,29), (7,31), (8,35), (9,37), (10,40), (11,43)$

Which line is the better fit (closer to more points overall)?

Line B, because it goes through the highest point exactly
Line A, because a best-fit line must pass through $(0,0)$
Line A, because it stays closer to the points for most $x$ -values (correct answer)
Line B, because it is steeper and steeper lines are always better

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of push-ups vs points scored showing upward linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points within ±2 units of line vertically (good fit), or comparing two possible lines where one passes closer to majority of points (better fit). The correct line is Line A, because it stays closer to the points for most x-values, with smaller vertical distances overall. A common error is choosing Line B because it is steeper, but steepness alone doesn't determine fit; the line must minimize deviations from the actual data points. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 14

A scatter plot shows the relationship between $x$ = number of laps run and $y$ = minutes of exercise. The points follow a clear linear trend.

Which description best explains how to improve a line of fit that is currently drawn too high (most points are below it)?

Move the line to pass through the single highest point, even if it misses the rest
Rotate the line until all points are below it
Make the line vertical so it hits more points
Shift the line downward so the points are more balanced above and below the line (correct answer)

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of laps run vs exercise minutes showing upward linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points within ±5 minutes of line vertically (good fit), or comparing two possible lines where one passes closer to majority of points (better fit). The correct improvement is to shift the line downward so the points are more balanced above and below the line, ensuring it goes through the middle of the point cloud. A common error is moving the line to pass through a single high point, but this ignores the overall balance and majority of data. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 15

A scatter plot shows (x = number of laps swum, y = total time in minutes). The points have a clear positive linear trend. A student drew the line $y=2x+1$ , but most points are above the line for all $x$ values. What is the main problem with the student's line?

The line is not centered on the data; it should be adjusted upward so points are more balanced above and below the line. (correct answer)
The line should slope downward, not upward.
The line is correct, because a best-fit line should have almost all points above it.
The line is too high; it should be shifted upward so most points are below it.

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of laps swum vs time showing positive trend, if most points are above the line y=2x+1, it indicates the line is too low and unbalanced, needing upward adjustment for better centering. The main problem is that the line is not centered on the data; it should be adjusted upward so points are more balanced above and below the line. Common errors include thinking the line should have all points on one side or wrong slope direction. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 16

A science class measured the temperature (x) and the number of crickets chirping per minute (y). The scatter plot points form a pattern that curves upward (not a straight trend). Should a straight line be used as a model?

Yes, because any scatter plot can be modeled well with a straight line.
No, because the pattern is curved, so a straight line would not match the trend well. (correct answer)
No, because the points are close together (that means you cannot model them).
Yes, because a straight line must pass through the first and last points.

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of temperature vs crickets chirping showing curved upward pattern, attempting to fit a straight line would leave many points far from it (poor fit), whereas a curved model would better capture the trend. The correct answer is no, because the pattern is curved, so a straight line would not match the trend well. Common errors include assuming any scatter can be linearly modeled or misjudging based on points being close together. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 17

A class collected data on study time and quiz score. The scatter plot shows a clear positive linear trend. Which line would be the best informal line of best fit (drawn by eye)? Choose the option that describes the best-fit line.

Data points (study hours, quiz score): $(1, 58), (2, 61), (2, 64), (3, 66), (3, 69), (4, 72), (4, 74), (5, 77), (5, 79), (6, 82), (6, 84), (7, 86)$

A horizontal line near $y=75$ because most scores are between 70 and 85
A line that passes through every point by zig-zagging slightly
A line that goes through the middle of the points, rising from left to right, with about the same number of points above and below it (correct answer)
A line with a negative slope because some scores go down between nearby points

Explanation: This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of study hours vs quiz scores showing upward linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points within ±5 points of line vertically (good fit), or comparing two possible lines where one passes closer to majority of points (better fit). The correct choice is a line that goes through the middle of the points, rising from left to right, with about the same number of points above and below it, as it balances the data and follows the positive trend. A common error is suggesting a line that passes through every point by zig-zagging, but scatter plots show trends, not exact fits, so straight lines approximate the pattern without hitting all points. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.

Question 18

Use the scatter plot shown to answer the question. Which of the following best describes why a straight line is NOT an appropriate model for these data?

The data points show a clear linear increase.
The data points form a curved (nonlinear) pattern. (correct answer)
There are exactly 10 data points shown.
The x-values are not evenly spaced.

Explanation: A straight line is an appropriate model only when the data suggest a linear association. If the data form a curve, a straight line will not fit well. Choice A contradicts the premise. Choice C is irrelevant — the number of points does not determine linearity. Choice D is also irrelevant; uneven x-spacing does not prevent a linear model.

Question 19

A scatter plot shows (x = number of chores completed, y = allowance earned). Two students drew different lines. Student 1's line has most points about 1-2 dollars away from it. Student 2's line has most points about 5-6 dollars away from it. Which line is the better fit?

Student 2's line
Both lines fit equally
Neither line fits
Student 1's line (correct answer)

Explanation: A good line of best fit has small vertical distances, or residuals, between the line and most of the data points. Student 1's line has residuals of only about 1-2 dollars, while Student 2's line has much larger residuals of about 5-6 dollars, so Student 1's line fits the data better. Choice A is wrong because Student 2's larger residuals mean the line is farther from the actual data, not more accurate. Choice B is wrong because the two lines don't fit equally well; one has noticeably smaller residuals than the other. Choice C is wrong because Student 1's line does fit reasonably well, given its small residuals.

Question 20

Based on the scatter plot shown, Maria draws a line to model the relationship between hours of study time and test scores. Which statement best evaluates how well her line fits the data?

The line fits well because it passes through exactly two data points and shows a clear upward trend.
The line fits poorly because several data points are more than 10 units away from the line vertically.
The line fits well because most data points are clustered close to the line with only minor deviations. (correct answer)
The line fits perfectly because it shows that more study time always results in higher test scores.

Explanation: A good line of fit should have most data points close to the line, with roughly equal numbers of points above and below. Choice C correctly identifies this criterion. Choice A is wrong because a line doesn't need to pass through exact points to fit well. Choice B is wrong because the specific distance mentioned (10 units) isn't necessarily poor fit without context of the data scale. Choice D is wrong because no real-world line fits perfectly, and correlation doesn't imply causation.