Fit Lines to Scatter Plots - 8th Grade Math

1. 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?
   Answer: The line is not centered on the data; it should be adjusted upward so points are more balanced above and below the line. 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.
2. Refer to the scatter plot below. Which equation best represents a line of fit for the data shown?
   Answer: $y = -2x + 20$ The data points show a clear negative trend, decreasing from about (1, 18) to (9, 2), which corresponds to a slope of about $-2$ and a y-intercept near 20. Choice A has positive slope, which contradicts the downward trend. Choice C also has positive slope. Choice D has the correct sign but too small a magnitude of slope and wrong y-intercept.
3. 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.
   Answer: About 42 minutes 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.
4. 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?
   Answer: Draw the line through the center of the main cluster of points, not letting the outlier control the line 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.
5. 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?
   Answer: Line A, because it goes through the middle of the point cloud and leaves about the same number of points above and below. 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.
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?
   Answer: Move the line downward so it goes through the middle of the points, with a more balanced number of points above and below. 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.
7. The table below shows the number of ice cream cones sold at a shop at different outdoor temperatures. Which linear equation best models the data?
   Answer: $y = 3x - 130$ The slope from (60, 50) to (90, 140) is $\frac{140-50}{90-60} = \frac{90}{30} = 3$, and using $y = 3x + b$ with (60, 50): $50 = 180 + b$, so $b = -130$. Choice A gives values way too small. Choice C has the correct slope but wrong intercept sign. Choice D underestimates both slope and produces mostly negative values in this range.
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?
   Answer: The slope is the wrong direction; the line should have a positive slope to match the upward trend. 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.
9. 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?
   Answer: The vertical distances from the points to the line (how far up or down each point is from the line). 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.
10. 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 $y=2x+5$?
    Answer: Good fit, because most points are within about 1–2 minutes vertically from the line 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.
11. 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): $(1, 58), (2, 61), (2, 64), (3, 66), (3, 69), (4, 72), (4, 74), (5, 77), (5, 79), (6, 82), (6, 84), (7, 86)$
    Answer: 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 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.
12. 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?
    Answer: Student 1's line 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.
13. Based on the scatter plot shown, three students draw different lines to model the data. Line R passes through the point with the highest y-value and the point with the lowest y-value. Line S passes through the center of the data with equal numbers of points above and below. Line T is horizontal through the mean of all y-values. Which line represents the best linear model?
    Answer: Line S, because it balances the data points and follows the overall linear trend most effectively. Line S represents the best linear model because it balances the data points above and below while following the overall trend. This is the principle behind fitting lines to data. Line R (connecting extreme points) ignores the pattern of most data points. Line T (horizontal through the mean) ignores any linear relationship that exists. Choice C incorrectly describes what a horizontal line accomplishes.
14. 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?
    Answer: The line fits well because most data points are clustered close to the line with only minor deviations. 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.
15. 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)?
    Answer: Line A, because it stays closer to the points for most $x$-values 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.
16. A line of best fit for a scatter plot of (x = minutes practiced, y = free throws made) is drawn as $y \approx 0.8x + 2$. Using this line, about how many free throws would be predicted for $x=20$ minutes?
    Answer: About 18 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 minutes practiced vs free throws made showing upward linear trend, using the line y ≈ 0.8x + 2 to predict for x=20 gives y ≈ 18, assessing fit by how well it approximates actual points. The correct prediction is about 18, calculated as 0.8*20 + 2 = 18, using the line for estimation. Common errors include miscalculating the prediction like choosing 16 (forgetting to add 2) or 22 (using wrong slope). 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.
17. 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?
    Answer: Good fit, because most points are close to the line (small vertical distances). 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.
18. 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)?
    Answer: Shift the line downward so the points are more balanced above and below the line 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.
19. 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?
    Answer: Check whether the points are vertically close to the line (small up-and-down distances) 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.
20. 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?
    Answer: Draw the line to fit the main cluster of points, without letting the one outlier determine the slope. 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.

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