Two-Variable Data: Models and Scatterplots Flashcards

A researcher fits a linear model to data and reports the best-fit line y = 3.2x + 7.5. The data range for x is 1 to 20. If someone uses the model to predict y at x = 100, what concern is raised?

Answer: Extrapolation — the model is being used outside its reliable data range.

Using a model beyond the range of the data (extrapolation) can yield unreliable predictions.

Two variables x and y have a correlation of r = −0.65. Which of the following is a correct interpretation?

Answer: There is a moderate negative linear relationship between x and y.

r = −0.65 indicates a moderate (not strong) negative linear correlation; it does not imply causation.

Data on water temperature (x, °C) and dissolved oxygen (y, mg/L) has the linear model y = −0.4x + 14. What dissolved oxygen level is predicted at 20°C?

Answer: 6 mg/L

y = −0.4(20)+14 = −8+14 = 6 mg/L.

A quadratic model y = x² − 8x + 12 is fit to data. For which x values does y = 0?

Answer: x = 2 and x = 6

Factor: x²−8x+12 = (x−2)(x−6) = 0 → x=2 or x=6.

A scatterplot of two variables shows points tightly clustered around a line with negative slope. Which range for r is most likely?

Answer: −1.0 to −0.9

Tight clustering around a line with negative slope indicates a strong negative linear correlation, so r is close to −1.

A model is described as y = 500 − 15x. After how many years (x) does y drop below 100?

Answer: More than 26 years

Set 100 > 500−15x → 15x > 400 → x > 26.67; so y drops below 100 after more than 26 years.

A table of x and y shows constant ratios between successive y values as x increases by 1. Which of the following model types best fits this data?

Answer: Exponential

Constant multiplicative ratios between successive y-values is the hallmark of an exponential model.