Sale price per square foot declines as homes get larger. Does that fact alone prove that a straight-line model of total price on area is inadequate?
Correct Answer
C) No; a straight line with a positive intercept can also produce declining unit price
If P = a + bA with a positive intercept a, then P/A = a/A + b falls as area A grows even though total price P is linear in A. Declining unit price alone therefore does not establish curvature. Inspect total-price plots, residuals and model assumptions rather than automatically removing area or choosing a transformation.
Why This Is the Correct Answer
Option C is correct because if total price P = a + bA with a positive intercept a, then price per square foot is P/A = a/A + b. As area A grows, a/A shrinks, so unit price falls even though total price is linear in area. Declining unit price alone therefore does not establish curvature.
Why the Other Options Are Wrong
Option A: Yes; a declining unit price proves the total-price curve must bend downward with size
Option A says a declining unit price proves the total-price curve must bend downward with size, but a straight line with a positive intercept also produces declining unit price. The decline alone does not prove curvature.
Option B: Yes; area must therefore be omitted as a predictor
Option B says area must therefore be omitted as a predictor, which does not follow. Declining unit price says nothing about removing area; area may still be strongly related to total price.
Option D: No; it instead proves that the observed sale prices are wrong
Option D says the decline proves the observed sale prices are wrong, but declining unit price is a normal and expected pattern. It does not indicate data errors.
Fixed Part Spreads Thinner
A positive intercept is a fixed amount spread over more square feet as size grows, so unit price falls even on a straight line.
How to use: Use this idea when a question links declining price per square foot to model curvature.
Exam Tip
Do not treat declining unit price as proof of nonlinearity; check the total-price relationship and residuals instead.
Common Mistakes to Avoid
- -Assuming declining price per square foot proves curvature
- -Dropping area from the model because unit price falls
- -Confusing unit-price patterns with total-price patterns
Concept Deep Dive
Analysis
This question tests whether declining price per square foot proves a straight-line model of total price is wrong. Working through the algebra shows it does not: a linear model with a positive intercept produces a unit price that falls as size increases. The evidence for curvature must come from total-price plots and residuals.
Background Knowledge
Price per square foot commonly decreases as homes get larger, partly because fixed components such as land and basic systems are spread over more area. A linear total-price model with a positive intercept captures this pattern, so model choice should rest on total-price plots, residual analysis and model assumptions.
Real-World Application
An appraiser notices price per square foot falls from $260 for 1,200-square-foot homes to $190 for 2,800-square-foot homes. A plot of total price against area looks straight with a positive intercept, so the appraiser keeps the linear model rather than adding an unnecessary transformation.
More Statistics Questions
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