Let's go through each statement carefully based on the context that function $f$ is a quadratic regression model, and function $g$ is another regression model (possibly linear or quadratic) that more accurately models the data.

A. Function f likely had fewer residuals near the x-axis than function g.

• False. The residuals of a regression model represent the differences between the observed data points and the model's predicted values. If function $f$ is less accurate than $g$, we would expect that the residuals for $f$ would be larger or more scattered. This doesn't necessarily mean there would be fewer residuals near the x-axis (where residual = 0), just that the model $f$ would likely have more dispersed residuals.

B. Function f likely had more residuals equal to 0 than function g.

• False. The residuals being equal to 0 would imply that the model perfectly predicts those points. Since model $g$ is described as more accurate, it's more likely that $g$ would have more residuals close to 0 than $f$.

C. Function g had more residuals near the y-axis than function f.

• False. Residuals near the y-axis correspond to predictions where the independent variable $x$ is near 0. The accuracy of a model doesn't necessarily have to do with the proximity of residuals to the y-axis, but rather how closely the model's predictions match the observed values across all points. Since $g$ is more accurate, it should generally have smaller residuals overall, not necessarily more near the y-axis.

D. Function g likely had an $R^2$ value closer to 1 than function f.

• True. The $R^2$ value indicates the proportion of the variance in the data that is explained by the model. A more accurate model typically has an $R^2$ value closer to 1. Since $g$ is said to more accurately model the data, it likely has a higher $R^2$ than $f$.

E. Function g likely had all positive residual values.

• False. Residuals can be both positive and negative depending on whether the model overestimates or underestimates the data at any point. It's unlikely that all residuals for $g$ would be positive because residuals usually vary in sign across different data points.

Conclusion:

The only true statement is:

• D. Function g likely had an $R^2$ value closer to 1 than function f.

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Further Questions:

1. How does the value of $R^2$ relate to the accuracy of a regression model?
2. What is the difference between residuals and errors in regression analysis?
3. Why might a quadratic regression model not always provide a better fit than other types of regression models?
4. How does the residual plot help in evaluating the fit of a regression model?
5. What factors can cause a regression model to have a lower $R^2$ value?

Tip:

When comparing regression models, it’s helpful to look at both the $R^2$ value and the residual plot. A higher $R^2$ typically indicates a better fit, but the residual plot can reveal patterns that suggest if the model is missing a key aspect of the data.