Question 51·Hard·Two-Variable Data: Models and Scatterplots
A tutoring company recorded the number of full-length practice tests each of 8 students completed and the increase in each student’s SAT Math score. The data are shown.
| Practice tests completed, | Score increase (points), |
|---|---|
| 1 | 15 |
| 2 | 30 |
| 2 | 28 |
| 3 | 43 |
| 4 | 46 |
| 5 | 58 |
| 6 | 71 |
| 7 | 80 |
According to the least-squares regression line for these data, which of the following is closest to the predicted average increase in score, in points, for each additional practice test completed?
On SAT questions that ask for the “predicted change in y for each additional x” or use phrases like “for each additional” or “per unit,” interpret this as the slope of the linear model. If the exact regression line is not given, quickly approximate the slope by choosing two points far apart along the trend, computing , and then matching that approximate value to the closest answer choice. This avoids heavy computation and uses the overall pattern of the data, which is all the test expects.
Hints
Identify the key quantity
Focus on the phrase “for each additional practice test”. In a linear model, what part of the equation tells you how much changes when increases by 1?
Connect the phrase to the regression line
The least-squares regression line is a line of best fit. The question is asking for the slope of this line. How can you approximate the slope from data points in a roughly linear pattern?
Choose helpful points
Pick two data points that are far apart in to estimate the slope: one with a small number of tests and one with a large number of tests. Compute for those points, then compare that value to the answer choices.
Desmos Guide
Enter the data as a table
Create a table in Desmos. In the first column (it will be labeled ), enter the practice tests: 1, 2, 2, 3, 4, 5, 6, 7. In the second column (labeled ), enter the score increases: 15, 30, 28, 43, 46, 58, 71, 80.
Fit a regression line
In a new expression line, type y1 ~ a x1 + b. Desmos will compute the least-squares regression line and display values for parameters and .
Interpret the regression output
Look at the value of in the regression equation; this is the slope of the best-fit line and represents the predicted average increase in score per additional practice test. Compare this value to the answer choices and select the one it is closest to.
Step-by-step Explanation
Understand what the question is really asking
The phrase “predicted average increase in score for each additional practice test” refers to the slope of the least-squares regression line.
- In a linear model , the slope tells you how much (score increase) is predicted to change when (practice tests) increases by 1.
- So we need to estimate the slope of the best-fit line for these data.
Estimate the slope from the data
The data show a roughly linear relationship, so we can approximate the slope using two points that are far apart and representative.
Use the first and last data points:
- Point 1:
- Point 2:
Compute the change in (score increase) and change in (tests):
So the approximate slope is
This is the predicted average increase in score (in points) for each additional practice test.
Match your estimate to the answer choices
Our estimate for the slope is about points per test.
Compare this to the choices: 4, 8, 10, and 14. The value 10 is closest to .
Correct answer: 10.