In the context of constructing confidence intervals, when is a linear model generally not a good fit for a set of data?

• A. When the sample size is large and the data are approximately normally distributed
• B. When the data points are closely clustered around a straight line
• C. When the residuals from the model are randomly scattered with constant variance
• D. When the relationship between the variables is clearly nonlinear, such as a curved or exponential pattern in the data

Answer: (D)

Verified step by step guidance

1. Understand that a linear model assumes a straight-line relationship between the independent and dependent variables.
2. Recognize that the appropriateness of a linear model depends on the pattern of the data and the residuals (differences between observed and predicted values).
3. Check the scatterplot of the data: if the data points show a clear curved or nonlinear pattern (e.g., exponential, quadratic), a linear model will not fit well.
4. Examine the residual plot: if residuals display a systematic pattern (not randomly scattered) or changing variance, this indicates the linear model is not appropriate.
5. Conclude that a linear model is generally not a good fit when the relationship between variables is clearly nonlinear, despite sample size or normality of data.