Two-Variable Data: Models and Scatterplots Flashcards

A scatterplot displays data with the line y = 7x − 3. A point at x = 5 has actual y = 38. Is the actual y above or below the line, and what is the residual?

Answer: Above the line; residual = +6

Predicted y = 7(5) − 3 = 32; actual = 38; residual = 38 − 32 = +6, so the point lies above the line.

If a linear model for x and y has slope 0 (a horizontal line), what does this imply about the correlation?

Answer: The correlation is 0; y does not change as x changes

A slope of 0 means y is constant regardless of x, implying no linear relationship, i.e., r = 0.

Data on the amount of fertilizer used (x) and crop yield (y) shows the model y = −0.1x² + 5x + 20. For what value of x is the yield maximized?

Answer: 25

Vertex at x = −b/(2a) = −5/(2·(−0.1)) = −5/(−0.2) = 25.

A company's revenue is modeled by y = 300(1.08)^x, where x is years since founding. What does 1.08 represent?

Answer: Revenue grows by 8% per year

A base of 1.08 means each year revenue is multiplied by 1.08, an 8% growth rate.

Two variables have a negative correlation. As x increases by 1, y changes by −4. The line of best fit has y-intercept 60. What is the equation?

Answer: y = −4x + 60

Slope = −4 and y-intercept = 60 gives y = −4x + 60.

A table shows x: 0, 1, 2, 3 and y: 5, 15, 45, 135. What is the growth factor of this exponential model?

Answer: 3

Each y is multiplied by 3 (15/5 = 3, 45/15 = 3, 135/45 = 3), so the growth factor is 3.

A line of best fit for a dataset has the equation y = 2.2x + 1.5. Which of the following is closest to the predicted y when x = 7.5?

Answer: 18

y = 2.2(7.5) + 1.5 = 16.5 + 1.5 = 18.