A study finds that as the number of firefighters at a fire ($x$) increases, so does the property damage ($y$), yielding $R^2 = 0.95$. What is the correct conclusion?

More firefighters cause more damage

The linear model fits the data well, but does not prove causation

The model is unreliable because $R^2$ is too high

Fewer firefighters should be sent to reduce damage

Which statement about the Coefficient of Determination $R^2$ is TRUE?

$R^2$ can be greater than 1 for a perfect fit

$R^2 = 0$ means the model perfectly fits the data

A value of $R^2$ close to 1 indicates the model fits the data well

A high $R^2$ always proves that $x$ causes $y$

10-8 Modeling and Curve Fitting Practice — Grade 9 math

10-8 Modeling and Curve Fitting — Practice Questions

1. A calculator runs three regressions on a dataset and returns: Linear $R^2 = 0.72$, Quadratic $R^2 = 0.91$, Exponential $R^2 = 0.88$. Which model best fits the data?
- A. Linear model
- B. Quadratic model
- C. Exponential model
- D. All models fit equally well
2. A study finds that as the number of firefighters at a fire ($x$) increases, so does the property damage ($y$), yielding $R^2 = 0.95$. What is the correct conclusion?
- A. More firefighters cause more damage
- B. The linear model fits the data well, but does not prove causation
- C. The model is unreliable because $R^2$ is too high
- D. Fewer firefighters should be sent to reduce damage
3. The $R^2$ values for three regression models are: LinReg $= 0.80$, QuadReg $= 0.76$, ExpReg $= 0.94$. The model that best fits the data is ___, and its $R^2$ value is ___.
4. Which statement about the Coefficient of Determination $R^2$ is TRUE?
- A. $R^2$ can be greater than 1 for a perfect fit
- B. $R^2 = 0$ means the model perfectly fits the data
- C. A value of $R^2$ close to 1 indicates the model fits the data well
- D. A high $R^2$ always proves that $x$ causes $y$
5. For the dataset $\{(1, 3), (2, 7), (3, 13), (4, 21), (5, 31)\}$, the second differences are all equal to $2$. Which regression type is most appropriate?
- A. LinReg
- B. QuadReg
- C. ExpReg
- D. CubicReg
6. A table of data has evenly spaced $x$-values. The first differences of the $y$-values are $5, 5, 5, 5$. Which function family best models this data?
- A. Linear
- B. Quadratic
- C. Exponential
- D. None of the above
7. Consider the data: $(0, 2),\ (1, 6),\ (2, 18),\ (3, 54)$. The successive ratios between consecutive $y$-values are all equal to ___.
8. For the data $(1, 2),\ (2, 5),\ (3, 10),\ (4, 17)$, the first differences are $3, 5, 7$. What are the second differences, and what function family does this suggest?
- A. Second differences are $2$; Quadratic
- B. Second differences are $2$; Linear
- C. Second differences are $3$; Exponential
- D. Second differences are $2$; Exponential
9. The data $(0, 5),\ (1, 8),\ (2, 11),\ (3, 14)$ has evenly spaced $x$-values. The constant first difference is ___.
10. A dataset has first differences of $2, 4, 8, 16$. Which method should you apply next to correctly identify the function family?
- A. Check if the second differences are constant
- B. Check if the successive ratios are constant
- C. Check if the first differences are constant
- D. No further analysis is needed