Which of the following has no logical interpretation in context regarding the regression line used to predict collar size from height?
Understand the Problem
The question is asking which of the given options has no logical interpretation in the context of the regression line that predicts collar size based on height for boys. It involves understanding the components of the regression equation provided.
Answer
E: The $c$-intercept of the regression line.
Answer for screen readers
The option that has no logical interpretation in context is E: The $c$-intercept of the regression line.
Steps to Solve
- Understand the Regression Equation
The regression line provided is $\hat{c} = -94 + 0.9h$, where $\hat{c}$ is the predicted collar size and $h$ is the height in centimeters. The equation shows a relationship where collar size is predicted based on height.
- Analyze Each Option
We need to evaluate each option and determine whether it logically fits within the context of predicting collar size based on height.
- Consider Option A: Predicted Collar Size for Height 140 cm
For a boy with height $h = 140$ cm, you can find the predicted collar size:
$$ \hat{c} = -94 + 0.9(140) $$
- Consider Option B: $h$ Values in the Sample
The $h$ values in the sample are simply the recorded heights of boys in the data. This has logical interpretation as it describes the independent variable.
- Consider Option C: $c$ Values in the Sample
The $c$ values (collar sizes) in the sample similarly represent the dependent variable values. This also has logical interpretation within the context of the relationship.
- Consider Option D: Slope of the Regression Line
The slope of $0.9$ represents the change in collar size for a unit change in height. This has a clear logical interpretation in the context of the regression analysis.
- Consider Option E: The $c$-intercept of the Regression Line
The $c$-intercept value $-94$ itself has a mathematical interpretation but does not have a logical interpretation in the context of boys' shirt collars as negative collar size does not make practical sense.
The option that has no logical interpretation in context is E: The $c$-intercept of the regression line.
More Information
The $c$-intercept indicates the estimated collar size when height is zero, which is not possible in this context as it does not correspond to any realistically measurable collar size for boys.
Tips
- Confusing the intercept's mathematical meaning with its practical relevance.
- Misinterpreting what is meant by a "logical interpretation" in reference to context.
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