25 hard SAT Two-variable data: models & scatterplots questions
Real questions from the SAT Climb bank, all at the hard difficulty tier. Pick an answer before you open the explanation. Every question tells you why the answer is right and why each wrong choice is tempting.
Math · Problem-Solving and Data Analysis~2 per testHard tier
Confuses residual sign (above vs. below the line).
Question 1Hard
The scatterplot shows the relationship between the number of items produced per day, x, and the total production cost in dollars, y, for 14 days at a manufacturing plant. The data points roughly follow the line y=8x+1200, with x ranging from 100 to 400. On a day when 250 items were produced, the actual production cost was $3150. What is the sign of the residual for this data point?
Show the answer and explanation
Why A is right
The predicted cost for 250 items is y=8(250) + 1200 = 3200 dollars. The actual cost was $3150, which is less than the predicted $3200. The residual is actual minus predicted: 3150 - 3200 = -50, which is negative.
Why the others are wrong
BThis incorrectly concludes that the actual cost exceeds the predicted cost, reversing the comparison.
CThis incorrectly relates the sign of the residual to the sign of the slope, which are independent.
DThis confuses whether the x-value is within the data range with the sign of the residual.
Question 2Hard
The scatterplot shows the relationship between temperature in degrees Celsius, x, and electricity usage in kilowatt-hours, y, for 6 days. The data points are approximately (15, 45), (18, 51), (21, 57), (24, 63), (27, 69), and (30, 75). A line of best fit has equation y=2x+c, where c is a constant. Based on the data, what is the value of c?
Show the answer and explanation
Why B is right
Using the point (15, 45): 45 = 2(15)+c, so 45=30+c, and c=15. This can be verified with any other point such as (24, 63): 63 = 2(24)+c gives c=15.
Why the others are wrong
AThis incorrectly uses the slope as the y-intercept.
CThis gives the y-value of the first data point rather than solving for the parameter c.
DThis gives the result of 2(15) rather than completing the calculation to find c.
Question 3Hard
A biologist studied the relationship between the depth, in meters, and the water temperature, in degrees Celsius, at various points in a lake. The data set contains the data from 12 measurements. The equation of the line of best fit is y=18−0.6x, where x is the depth, in meters, and y is the temperature, in degrees Celsius.
Based on the line of best fit, what is the predicted decrease in temperature, in degrees Celsius, for each increase of 5 meters in depth?
Show the answer and explanation
Why B is right
The slope of the line is -0.6, which means the temperature decreases by 0.6 degrees Celsius for each 1-meter increase in depth. For a 5-meter increase in depth, the temperature decrease is 0.6 × 5 = 3 degrees Celsius.
Why the others are wrong
AThis is the rate of change per 1 meter of depth, not per 5 meters. The student must multiply this rate by 5 to find the change over 5 meters.
CThis results from incorrectly adding the rate to the depth change (0.6 + 3 = 3.6) rather than multiplying the rate by the depth change.
DThis is the y-intercept of the model, representing the predicted temperature at depth 0, not the change in temperature over a 5-meter depth increase.
Question 4Hard
A hydrologist measured the water level in a reservoir over a 60-day period. The water level remained approximately 85 meters for the first 40 days, then rose rapidly due to heavy rainfall. Which of the following types of functions would best model the relationship between day number and water level for the entire data set?
Show the answer and explanation
Why B is right
The water level data show two distinct phases: constant level for 40 days, then rapid rise. Only a piecewise function can accurately model this by combining a constant function for the initial period with a growth function for the rainfall period.
Why the others are wrong
AA quadratic function would show continuous curvature throughout the 60 days, failing to represent the distinct flat-then-rising pattern with an abrupt transition.
CA linear function would show constant rate of increase from day 1, missing the 40-day period of constant water level.
DAn exponential function would show water level growth from the beginning, unable to capture the initial stable 40-day period.
Question 5Hard
The scatterplot shows the relationship between study time in hours, x, and exam scores, y, for 20 students. A line of best fit for the data passes through the points (3, 68) and (7, 84). A student who studied for 5 hours scored 72 on the exam. What is the residual for this student?
Show the answer and explanation
Why A is right
The slope is (84 - 68)/(7 - 3) = 16/4 = 4. Using point (3, 68): y−68=4(x−3). When x=5: y−68=4(2) = 8, so y=76. The residual is 72 - 76 = -4.
Why the others are wrong
BThis results from computing predicted minus actual (76 - 72 = 4) instead of actual minus predicted.
CThis results from reporting the predicted value 76 instead of the residual.
DThis results from doubling the actual residual, perhaps from misapplying the slope in the residual calculation.
Question 6Hard
A scatterplot shows the relationship between the number of hours studied, x, and test score, y, for 15 students. The data points lie approximately along the line y=3.2x+42, with x ranging from 5 to 20 hours. One student studied for 12 hours and scored 79 points.
The scatterplot shows the relationship between the number of hours studied and test score for 15 students. Based on the line of best fit for the data, what is the residual for the student who studied for 12 hours?
Show the answer and explanation
Why A is right
The line of best fit is y=3.2x+42. For x=12, the predicted value is y=3.2(12) + 42 = 38.4 + 42 = 80.4. The residual is actual minus predicted: 79 - 80.4 = -1.4.
Why the others are wrong
BThis is the absolute value of the residual, incorrectly treating it as predicted minus actual instead of actual minus predicted.
CThis incorrectly uses the slope value as the residual without performing the calculation.
DThis uses the predicted value from the line of best fit rather than computing the residual.
Question 7Hard
The scatterplot shows the relationship between the number of hours of sleep, x, and reaction time in milliseconds, y, for 8 participants. The data points are approximately (4, 320), (5, 300), (6, 280), (7, 260), (8, 240), (9, 220), (10, 200), and (11, 180). A line of best fit has slope -20 and passes through the point (7, 260). What is the equation of the line of best fit?
Show the answer and explanation
Why C is right
Using the point-slope form with slope -20 and point (7, 260): 260 = -20(7)+b, so 260=−140+b, and b=400. The equation is y=−20x+400.
Why the others are wrong
AThis incorrectly uses the y-value of the given point as the y-intercept without solving for b.
BThis uses -20(7) = -140 as the y-intercept without completing the calculation.
DThis incorrectly uses a positive slope, ignoring that reaction time decreases with more sleep.
Question 8Hard
A meteorologist recorded atmospheric pressure over a 24-hour period. The pressure remained approximately 1013 millibars for the first 16 hours, then decreased rapidly. Which of the following types of functions would best model the relationship between time and atmospheric pressure for the entire data set?
Show the answer and explanation
Why C is right
The pressure data exhibit two distinct phases: stability for 16 hours followed by rapid decrease. A piecewise function can model this by combining a constant function for the stable period with a decreasing function for the later period.
Why the others are wrong
AAn exponential function would show continuous decay from the beginning, failing to capture the initial 16-hour stable period.
BA quadratic function would show smooth curvature throughout the 24 hours, unable to represent the abrupt change from constant to rapidly decreasing pressure.
DA linear function would show constant rate of decrease from hour zero, missing the initial stable period entirely.
Question 9Hard
A scatterplot shows the relationship between the number of years of experience, x, and annual salary, y thousand dollars, for 22 employees. The data points approximately follow the line y=3.5x+45, with x ranging from 0 to 18. An employee with m years of experience has an annual salary of 80 thousand dollars, and this point lies on the line of best fit.
What is the value of m?
Show the answer and explanation
Why B is right
Since the point (m, 80) lies on the line y=3.5x+45, substituting y=80 gives 80=3.5m+45, so 35=3.5m and m=10.
Why the others are wrong
AThis incorrectly uses a slope of 5 instead of 3.5 when solving for m.
CThis incorrectly solves 80 = 3.5m without accounting for the y-intercept of 45.
DThis gives the value of 80 - 45 = 35 without dividing by the slope.
Question 10Hard
The scatterplot shows the relationship between the distance from a city center, x, in miles, and average home prices, y, in thousands of dollars, for 16 neighborhoods. The data points roughly follow the line y=−12x+480, with x ranging from 5 to 25. A neighborhood located 15 miles from the city center has an average home price of $330000. What is the residual for this neighborhood, in thousands of dollars?
Show the answer and explanation
Why C is right
The predicted home price for 15 miles is y=−12(15) + 480 = -180 + 480 = 300 thousand dollars. The actual price is $330000, or 330 thousand dollars. The residual is 330 - 300 = 30 thousand dollars.
Why the others are wrong
AThis results from calculating predicted minus actual instead of actual minus predicted.
BThis uses the distance value instead of computing the residual.
DThis gives the predicted value from the line rather than the difference between actual and predicted.
Question 11Hard
A study recorded the relationship between the number of days after planting, x, and the height of a plant in centimeters, y. There are 7 data points with coordinates approximately (5,8), (10,13), (15,18), (20,23), (25,28), (30,33), and (35,38). A line of best fit is given by y=kx+3, where k is a constant. What is the value of k?
Show the answer and explanation
Why D is right
Using any point on the line, such as (5, 8): 8=k(5) + 3, so 5k=5, and k=1. This can be verified with other points like (20, 23): 23=k(20) + 3 gives k=1.
Why the others are wrong
AThis gives the y-intercept rather than solving for the slope parameter k.
BThis incorrectly uses the x-value of the first data point as the slope.
CThis computes an approximate average rate but makes an arithmetic error or uses endpoints incorrectly.
Question 12Hard
A researcher collected data on the temperature of a chemical reaction over time. The data show that the temperature remained constant at approximately 20°C for the first 10 minutes, then increased rapidly. Which of the following types of functions would best model the relationship between time and temperature for the entire data set?
Show the answer and explanation
Why D is right
The data exhibit two distinct regimes: constant temperature for the first 10 minutes, then rapid increase. A piecewise function is the only model type that can capture this abrupt transition between a horizontal segment and a growth segment.
Why the others are wrong
AA linear model would suggest constant growth throughout, missing the flat initial period entirely.
BA quadratic model would show smooth curvature throughout, unable to represent the flat initial segment followed by the sharp transition to rapid growth.
CAn exponential model would show growth from the beginning, failing to capture the initial constant-temperature period.
Question 13Hard
A study recorded the relationship between the age of a car in years, x, and its resale value in thousands of dollars, y, for 16 cars. A line of best fit for the data has a y-intercept of 35 and passes through the point (6, 20). According to the line of best fit, what is the predicted resale value, in thousands of dollars, of a car that is 9 years old?
Show the answer and explanation
Why A is right
The line passes through (0, 35) and (6, 20), so the slope is (20 - 35)/(6 - 0) = -15/6 = -2.5. The equation is y=−2.5x+35. When x=9: y=−2.5(9) + 35 = -22.5 + 35 = 12.5.
Why the others are wrong
BThis results from incorrectly computing the slope as positive 2.5 instead of negative 2.5, giving y = 2.5(9) + 35 = 22.5, then subtracting 5.
CThis results from using only the y-intercept adjustment without properly accounting for the age of the car, computing 35 - 6 - 1.5 or similar error.
DThis results from over-extrapolating the rate of decline, perhaps doubling the slope to -5, giving -5(9) + 35 + 10 through calculation error.
Question 14Hard
The scatterplot shows the relationship between advertising spending, x, in thousands of dollars, and monthly revenue, y, in thousands of dollars, for 22 months. The points approximately follow the line y=5x+120, where x ranges from 10 to 40. If the company spent 35 thousand dollars on advertising in a month and earned 295 thousand dollars in revenue, what is the residual for this month, in thousands of dollars?
Show the answer and explanation
Why D is right
The predicted revenue at x=35 is y=5(35) + 120 = 175 + 120 = 295 thousand dollars. The actual revenue is also 295 thousand dollars, so the residual is 295 - 295 = 0.
Why the others are wrong
AThis incorrectly calculates a residual using a wrong predicted value, possibly from misapplying the slope.
BThis uses an incorrect predicted value with the wrong sign for the residual calculation.
CThis uses the actual y-value without subtracting the predicted value.
Question 15Hard
A study recorded the relationship between the age of a car in years, x, and its resale value in thousands of dollars, y. There are 9 data points with coordinates approximately (1,22), (2,20), (3,18), (4,16), (5,14), (6,12), (7,10), (8,8), and (9,6). A line of best fit for the data has equation y=mx+24, where m is a constant. If the point (3,18) lies on the line of best fit, what is the value of m?
Show the answer and explanation
Why C is right
Substituting the point (3,18) into y=mx+24 gives 18=m(3)+24. Solving for m: 18=3m+24, so 3m=−6, and m=−2.
Why the others are wrong
AThis gives the total change in y from x = 0 to x = 3 rather than the rate of change per unit of x.
BThis incorrectly uses the positive value of the slope, ignoring that the relationship shows decreasing resale value.
DThis gives the y-value of the point rather than solving for the slope parameter m.
Question 16Hard
An economist studied the price of a commodity over a 20-year period. The price remained stable at approximately $50 per unit for the first 12 years, then increased sharply. Which of the following types of functions would best model the relationship between time and price for the entire data set?
Show the answer and explanation
Why D is right
The data exhibit two distinct phases: stable pricing for 12 years followed by sharp increases. Only a piecewise function can model this by representing the initial stability with a constant segment and the later sharp increase with a different function.
Why the others are wrong
AA quadratic function would show continuous curvature throughout the period, failing to capture the distinct flat-then-rising pattern.
BA linear function would show constant rate of change from the beginning, missing the initial stable period.
CAn exponential function would show growth from year zero, unable to represent the 12-year stable period.
Question 17Hard
A study recorded the relationship between two variables, x and y. The data set contains 15 data points that follow an approximately linear pattern. A line of best fit for the data has a slope of 3.2 and passes through the point (5, 28). According to the line of best fit, what is the predicted value of y when x=12?
Show the answer and explanation
Why A is right
The line of best fit has slope 3.2 and passes through (5, 28). Using point-slope form: y−28=3.2(x−5). When x=12: y−28=3.2(7) = 22.4, so y=50.4.
Why the others are wrong
BThis results from incorrectly using the x-value 12 as the change in x from zero, computing 3.2(12) instead of accounting for the line passing through (5, 28).
CThis results from reversing the role of x and y in the point-slope calculation, treating 12 as a y-coordinate.
DThis results from incorrectly extrapolating by using a doubled slope value, computing as if the rate of change increases beyond the observed data range.
Question 18Hard
The scatterplot shows the relationship between years of experience, x, and annual salary, y, in thousands of dollars, for 30 employees. The data points lie approximately along the line y=6x+45, where x ranges from 0 to 20. For an employee with 12 years of experience earning 125 thousand dollars, is the residual positive or negative, and what does this indicate?
Show the answer and explanation
Why A is right
The predicted salary at x=12 is y=6(12) + 45 = 72 + 45 = 117 thousand dollars. The actual salary is 125 thousand dollars, which is above the predicted value. The residual is 125 - 117 = 8, which is positive, indicating the employee earns more than predicted.
Why the others are wrong
BThis incorrectly determines the sign of the residual; the actual salary of 125 is above the predicted 117, not below.
CThe sign of the residual depends on the position of the actual point relative to the line, not on the slope of the line itself.
DThe value x = 12 is within the given range of 0 to 20, so extrapolation is not an issue here.
Question 19Hard
A study recorded the relationship between the number of employees, x, and annual profit, y thousand dollars, for 19 companies. The data range from x=10 to x=200. A line of best fit has equation y=2.5x+b, where b is a constant. If the predicted profit for a company with 80 employees is 260 thousand dollars, what is the value of b?
Show the answer and explanation
Why B is right
Substituting x=80 and y=260 into y=2.5x+b gives 260 = 2.5(80)+b, so 260=200+b, which means b=60.
Why the others are wrong
AThis results from computing only 2.5(80) = 200 and stopping, without solving for b by subtracting from 260.
CThis results from misreading the x-value (80) as the answer, or from confusing the given values in the equation.
DThis results from incorrectly adding 260 + 200 instead of subtracting to solve for b.
Question 20Hard
A biologist recorded the population of bacteria in a culture. For the first 5 hours, the population remained approximately 200 bacteria. After 5 hours, the population began increasing rapidly. Which of the following types of functions would best model the relationship between time and population for the entire data set?
Show the answer and explanation
Why B is right
The data show two distinct behaviors: a constant population for the first 5 hours, then rapid growth. A piecewise function can model this by combining a constant function for the initial period with a growth function for the later period.
Why the others are wrong
AAn exponential function would show growth throughout the entire time period, failing to capture the initial constant population phase.
CA linear function would show constant growth from the start, missing the initial plateau entirely.
DA quadratic function would show smooth acceleration throughout, unable to represent the abrupt change from constant to growing population.
Question 21Hard
A study recorded the relationship between the number of hours studied, x, and test score, y, for 18 students. The line of best fit for the data is y=3.8x+52. Based on the line of best fit, what is the predicted increase in test score for each additional hour studied?
Show the answer and explanation
Why D is right
The slope of the line of best fit represents the predicted change in y for each one-unit increase in x. Since the equation is y=3.8x+52, the slope is 3.8, meaning the test score increases by 3.8 points for each additional hour studied.
Why the others are wrong
AThis incorrectly identifies the y-intercept as the rate of change.
BThis incorrectly calculates the predicted score at x = 1 rather than the rate of change.
CThis incorrectly multiplies the slope by the y-intercept.
Question 22Hard
The scatterplot shows the relationship between the age of a car, x, in years, and its resale value, y, in thousands of dollars, for 18 cars. The points approximately follow the line y=−2.5x+35, where x ranges from 1 to 10. For a car that is 6 years old with a resale value of 18 thousand dollars, is the residual positive or negative?
Show the answer and explanation
Why B is right
The predicted value at x=6 is y=−2.5(6) + 35 = -15 + 35 = 20 thousand dollars. The actual value is 18 thousand dollars, which is below the predicted value. Therefore, the residual (actual minus predicted) is 18 - 20 = -2, which is negative.
Why the others are wrong
AThis reverses the relationship; the actual value of 18 is below the predicted value of 20, not above it.
CThe sign of the residual depends on whether the actual value is above or below the line, not on the sign of the slope.
DExtrapolation beyond the data range is irrelevant to determining the residual for a point within the data range.
Question 23Hard
A researcher collected data on the age in years, x, and the value in thousands of dollars, y, of 20 used cars. A scatterplot of the data shows points that approximately follow the line y=−1.8x+24, with x ranging from 2 to 12 years. One car that is 7 years old has a value of 11 thousand dollars.
The scatterplot shows the relationship between the age and value of used cars. What is the average rate of change, in thousands of dollars per year, in the predicted value of a car from age 4 years to age 9 years?
Show the answer and explanation
Why B is right
The line of best fit is y=−1.8x+24. At x=4, y=−1.8(4) + 24 = 16.8. At x=9, y=−1.8(9) + 24 = 7.8. The average rate of change is (7.8 - 16.8)/(9 - 4) = -9.0/5 = -1.8 thousands of dollars per year.
Why the others are wrong
AThis incorrectly computes only the change in y-values (7.8 - 16.8 = -9.0) without dividing by the change in x-values.
CThis is the absolute value of the slope, incorrectly interpreting the negative rate of change as positive.
DThis uses the predicted value at x = 9 instead of computing the rate of change over the interval.
Question 24Hard
The scatterplot shows the relationship between the distance from city center, x, in miles, and monthly rent, y, in hundreds of dollars, for 24 apartments. The points approximately follow the line y=−3x+28, where x ranges from 1 to 8. What is the average rate of change in monthly rent, in hundreds of dollars per mile, from 3 miles to 7 miles from the city center?
Show the answer and explanation
Why C is right
The average rate of change is the slope of the line. At x=3, y=−3(3) + 28 = 19. At x=7, y=−3(7) + 28 = 7. The rate of change is (7 - 19)/(7 - 3) = -12/4 = -3 hundreds of dollars per mile, which matches the slope of the line.
Why the others are wrong
AThis calculates the total change in y-values (-12) without dividing by the change in x-values (4).
BThis gives the absolute value of the slope with the wrong sign, ignoring that rent decreases with distance.
DThis uses an incorrect interval or miscalculates the change in y divided by change in x.
Question 25Hard
The scatterplot shows the relationship between the number of hours of sleep, x, and reaction time, y, in milliseconds, for 16 participants. The data points roughly follow the line y=−15x+350, where x ranges from 4 to 10. What is the average rate of change in reaction time, in milliseconds per hour, from 5 hours of sleep to 8 hours of sleep?
Show the answer and explanation
Why A is right
The average rate of change is the slope of the line connecting the two points. At x=5, y=−15(5) + 350 = 275. At x=8, y=−15(8) + 350 = 230. The rate of change is (230 - 275)/(8 - 5) = -45/3 = -15 milliseconds per hour.
Why the others are wrong
BThis gives the absolute value of the slope, ignoring the negative sign that indicates reaction time decreases with more sleep.
CThis calculates the total change in y-values (-45) without dividing by the change in x-values (3).
DThis incorrectly doubles the slope or uses a wrong interval for the calculation.
These 25 are a sample, not a study plan
These 25 come from a bank of 17,599 questions across all 31 SAT skills and three difficulty tiers. Inside SAT Climb you get the rest of Two-variable data: models & scatterplots, a diagnostic that finds which skills are actually costing you points, and a grid that shows what to drill next.
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