ACT Math study guide · Chapter 33
Two-Variable Data and Models
A scatterplot shows the relationship between two numerical variables. A line of best fit, y = mx + b, summarizes that relationship and predicts a y-value for a given x-value. The slope m is the predicted change in y for each one-unit increase in x; the y-intercept b is the predicted value when x = 0.
A residual is the difference between an actual value and the predicted value. Residual = actual − predicted, not the reverse. A residual close to zero means the line of best fit predicted that point well; a large residual means the actual value fell far from what the model expected.
Correlation describes both the direction and the strength of a linear relationship. Points trending upward show positive correlation; points trending downward show negative correlation. Tightly clustered points along the trend indicate a strong correlation; loosely scattered points indicate a weak one. A strong correlation between two variables never proves that one causes the other; a plausible third variable can drive both at once.
A linear model is most appropriate when the scatterplot follows a roughly straight-line pattern. Clear curvature is a warning that a line may miss the relationship even when the variables are strongly associated.
▸ Predicted values and residuals
Worked Example A line of best fit is y = 4x + 3. An actual data point is (5, 28). Find its residual.
Find the predicted value first, then subtract it from the actual value.
Predicted value: 4(5) + 3 = 23
Residual = actual − predicted = 28 − 23 = 5
▸ Correlation, causation, and a third variable
When two variables show a strong correlation, three explanations are possible: one variable causes the other, the relationship runs in the opposite direction, or a third variable influences both. Without a randomized experiment, correlation alone cannot distinguish between these three explanations.
▸ Practice
1. A line of best fit is y = 3x − 4. Predict y when x = 6.
A) 18 B) 10 C) 22 D) 14
2. A line of best fit is y = 2x + 5. An actual data point is (6, 20). Find its residual.
F) 17 G) 3 H) −3 J) 20
3. A scatterplot shows points that trend downward from left to right, tightly clustered around a line. This describes what kind of correlation?
A) Strong negative B) Strong positive C) Weak positive D) No correlation
4. A study finds a strong positive correlation between ice cream sales and drowning incidents. What is the most likely explanation?
F) Ice cream sales cause drowning G) Drowning causes ice cream sales H) A third variable, such as hot weather, drives both J) The correlation is a calculation error
5. A line of best fit for hours studied versus test score is y = 4x + 60. What does the slope represent?
A) The test score with zero hours studied B) The number of hours studied C) The total number of students D) The predicted increase in test score per additional hour studied
6. A scatterplot shows almost no pattern, with points scattered evenly across the graph. This describes what kind of correlation?
F) Correlation near zero G) Strong positive H) Strong negative J) Perfect correlation
7. A line of best fit is y = 5x − 12. Predict y when x = 11.
A) 55 B) 38 C) 43 D) 48
8. A line of best fit is y = 2x + 9. An actual data point is (7, 27). Find its residual.
F) 4 G) −4 H) 23 J) 27
9. A scatterplot shows points loosely scattered with a slight downward trend. This describes what kind of correlation?
A) Strong positive B) Weak negative C) Strong negative D) No correlation
10. Cities with more libraries tend to have higher literacy rates. What is the most likely explanation?
F) Libraries directly cause literacy with no other explanation needed G) Literacy causes libraries to be built H) The correlation is a coincidence with no explanation J) A third variable, such as regional investment in education, likely drives both
11. A line of best fit for study hours versus test score is y = 6x + 50. What does the slope represent?
A) The predicted increase in score per additional hour studied B) The score with zero hours studied C) The total hours studied D) The total number of students
12. A scatterplot shows points clustered tightly along an upward line. This describes what kind of correlation?
F) Weak positive G) Strong positive H) Strong negative J) No correlation
13. A line of best fit for delivery cost versus distance is y = 3x + 8. What does the y-intercept represent?
A) The base cost when distance is zero B) The cost per mile C) The total distance D) The maximum possible cost
14. A scatterplot shows points with a very slight upward trend, loosely scattered. This describes what kind of correlation?
F) Strong positive G) No correlation H) Strong negative J) Weak positive
15. A line of best fit is y = 3x + 7. Predict y when x = 8.
A) 24 B) 31 C) 28 D) 35
16. A line of best fit is y = 4x − 2. An actual data point is (9, 40). Find its residual.
F) −6 G) 6 H) 34 J) 40
17. A scatterplot shows points clustered tightly along an upward line. Describe the correlation.
A) Strong negative B) Weak positive C) Strong positive D) No correlation
18. Cities with more ice cream shops tend to have higher sunburn rates. What is the most likely explanation?
F) A third variable, such as warm weather, likely drives both G) Ice cream causes sunburn directly H) Sunburn causes people to open ice cream shops J) The correlation is pure coincidence with no explanation
19. A line of best fit for advertising spend versus sales is y = 8x + 200. What does the slope represent?
A) Sales with zero ad spend B) The predicted increase in sales per additional unit of ad spend C) The total ad spend D) The total number of sales
20. A line of best fit for tree height (in cm) versus years since planting is y = 2x + 5. What does the y-intercept represent?
F) The growth rate per year G) The total number of years H) The tree's height at the time it was planted J) The maximum possible height
▸ Answers and Explanations
1: D. 3(6) − 4 = 18 − 4 = 14. Choice A reports 3(6) directly, forgetting to subtract 4.
2: G. Predicted value at x=6 is 2(6)+5=17; residual = actual − predicted = 20 − 17 = 3. Choice F reports the predicted value itself instead of the residual, which is the actual value minus the predicted value.
3: A. A downward trend means negative correlation; tight clustering around the line means the correlation is strong. Choice B mistakes the downward trend for an upward one, reversing the correlation's direction.
4: H. Hot weather plausibly increases both ice cream sales and swimming, without either variable causing the other. Choice F assumes a direct causal link between the two measured variables instead of considering a plausible third variable.
5: D. In a line of best fit written as y = mx + b, the slope m represents the predicted change in y for each one-unit increase in x, here each additional hour studied. Choice A confuses the slope with the y-intercept, which represents the value when x=0.
6: F. Points scattered with no visible trend indicate little to no linear relationship, a correlation near zero. Choice G mistakes a scattered pattern with no visible trend for a strong relationship.
7: C. 5(11)−12=55−12=43. Choice A reports 5(11) directly, forgetting to subtract 12.
8: F. Predicted value: 2(7)+9=23; residual=actual−predicted=27−23=4. Choice G drops the sign while computing the residual, subtracting the actual value from the predicted value instead of the reverse.
9: B. A downward trend means negative correlation; loose scattering means the correlation is weak. Choice A mistakes both the direction and the strength of the trend, reporting the opposite of both.
10: J. A plausible third variable, regional investment in education, could increase both the number of libraries and literacy rates, without either directly causing the other. Choice F assumes a direct causal link between the two measured variables instead of considering a plausible third variable.
11: A. In y=mx+b, the slope m represents the predicted change in y for each one-unit increase in x, here each additional hour studied. Choice B confuses the slope with the y-intercept, which represents the value when x=0.
12: G. An upward trend means positive correlation; tight clustering means the correlation is strong. Choice F correctly identifies the direction but misjudges the tight clustering as weak instead of strong.
13: A. In y=mx+b, the y-intercept b represents the predicted value when x=0, here the base cost before any distance is traveled. Choice B confuses the y-intercept with the slope, which represents the cost per mile.
14: J. A slight upward trend means positive correlation; loose scattering means the correlation is weak. Choice F correctly identifies the direction but misjudges the loose scattering as strong instead of weak.
15: B. 3(8)+7=24+7=31. Choice A reports 3(8) directly, forgetting to add 7.
16: G. Predicted value: 4(9)−2=34; residual=actual−predicted=40−34=6. Choice F drops the sign while computing the residual, subtracting the actual value from the predicted value instead of the reverse.
17: C. An upward trend means positive correlation; tight clustering means the correlation is strong. Choice A mistakes the upward trend for a downward one, reversing the correlation's direction.
18: F. A plausible third variable, warm weather, could increase both the number of ice cream shops and sunburn rates, without either directly causing the other. Choice G assumes a direct causal link between the two measured variables instead of considering a plausible third variable.
19: B. In y=mx+b, the slope m represents the predicted change in y for each one-unit increase in x, here each additional unit of ad spend. Choice A confuses the slope with the y-intercept, which represents the value when x=0.
20: H. In y=mx+b, the y-intercept b represents the predicted value when x=0, here the tree's height at the moment it was planted. Choice F confuses the y-intercept with the slope, which represents the growth rate per year.
▸ Evaluating a model
A model is not automatically good just because it produces a number. Check whether its predictions match the pattern in the data, whether residuals are randomly scattered around zero, whether the model is being used inside the range where the data were actually collected, and whether its parameters make sense in context. A systematic pattern in the residuals, instead of random scatter, means the model has missed real structure in the data.
1. Two models are fit to the same data. Model A's residuals are randomly scattered around 0. Model B's residuals form a clear curve, starting positive, dipping negative, then turning positive again. Which model fits better?
A) Model A B) Model B C) Both fit equally well D) Neither model can be evaluated from residuals
2. A line of best fit is built from data collected between x = 10 and x = 50. Using this line to predict a value at x = 200 is an example of:
F) Interpolation, which is always reliable G) Extrapolation, which carries more risk since it's outside the observed data range H) A residual calculation J) A correlation coefficient
3. A model predicts a person's height based on age, using data only from ages 5 to 15. The model predicts a 40-year-old is 9 feet tall. What does this best illustrate?
A) The model correctly extended a reliable trend B) The model was extrapolated far outside its data range, producing an unreasonable result C) The correlation coefficient must have been negative D) The residuals must be randomly scattered
▸ Answers and Explanations
1: A. Randomly scattered residuals suggest a model has captured the real pattern in the data, leaving only unpredictable noise. A curved residual pattern, as in Model B, signals that the model is missing real structure, often meaning a different type of model would fit better. Choice B selects the model whose residual pattern is actually a warning sign.
2: G. Interpolation predicts within the range of the observed data; extrapolation predicts outside that range, where the model's relationship is no longer confirmed by any actual data. Choice F claims interpolation is what's happening and calls it universally reliable, when x=200 falls outside the model's observed range entirely.
3: B. Predicting a 40-year-old's height from a model built only on ages 5 to 15 extends far beyond the data the model was built from, and the absurd result (9 feet) is the signature of unreliable extrapolation. Choice A treats this extreme extrapolation as if it were a safe continuation of a confirmed trend.