An intercept OLS model of residential sale price reports residual standard error $24,500, computed as sqrt(SSE/(n−p)), where p counts fitted coefficients. What is the most direct interpretation?
Correct Answer
B) It estimates the dollar scale of residual error under the model assumptions
The residual standard error is sqrt(SSE/(n−p)), with p counting fitted coefficients including the intercept in a full-rank OLS model. Here it estimates the residual error scale at $24,500 under the model assumptions. It is neither the mean signed residual (which is zero in intercept OLS) nor each slope's standard error, and it does not guarantee that every future price falls within $24,500 of prediction. Individual prediction intervals additionally reflect parameter-estimation uncertainty and predictor settings.
Why This Is the Correct Answer
Option B is correct because the residual standard error, sqrt(SSE/(n − p)), estimates the dollar scale of the model's residual error under the model assumptions, here $24,500. It summarizes how far sale prices typically fall from the fitted values.
Why the Other Options Are Wrong
Option A: The mean signed residual must equal $24,500
Option A says the mean signed residual must equal $24,500, but in an OLS model with an intercept the mean signed residual is zero. The residual standard error measures spread, not the average signed error.
Option C: Each future sale price will fall within $24,500 of the model’s prediction
Option C says each future sale price will fall within $24,500 of the prediction, but the residual standard error does not guarantee any such bound. Prediction intervals also reflect parameter-estimation uncertainty and predictor settings.
Option D: Each fitted slope has a standard error of $24,500
Option D says each fitted slope has a standard error of $24,500, but coefficient standard errors are separate figures that depend on the data for each predictor. The residual standard error describes the model's overall error scale.
Typical miss in dollars
Residual standard error is the model's typical miss, measured in the same dollars as sale price.
How to use: Use this to interpret residual standard error without confusing it with other regression outputs.
Exam Tip
Be cautious of answers promising that every future sale will fall within a fixed amount; regression statistics describe typical error, not guarantees.
Common Mistakes to Avoid
- -Treating residual standard error as a guaranteed prediction band
- -Confusing it with coefficient standard errors
- -Thinking the average residual equals the residual standard error
Concept Deep Dive
Analysis
This question tests interpreting the residual standard error of an OLS regression. It is the square root of the sum of squared errors divided by the residual degrees of freedom, and it estimates the typical size of the model's errors in dollars. The candidate must avoid confusing it with the mean residual, a guaranteed prediction band or coefficient standard errors.
Background Knowledge
In OLS regression, the residual standard error is the square root of the estimated error variance, using n − p as the divisor, where p counts fitted coefficients including the intercept. It is expressed in the units of the dependent variable. Smaller values indicate fitted prices closer to actual prices, subject to the model's assumptions.
Real-World Application
An appraiser using a regression model to support adjustments sees a residual standard error of $24,500. She explains in her report that the model's estimates typically miss actual prices by roughly that scale, and she does not present it as a guaranteed accuracy range.
More Statistics Questions
Sale prices are $270,000, $290,000, $300,000, $310,000, $330,000. Treat the listed observations as the complete population and divide by n when computing the standard deviation. Report to the nearest dollar.
A sales sampling frame systematically excludes a relevant part of the target market. What is the principal risk?
An appraiser estimates a market-wide price trend using only sales data from one brokerage that specializes in luxury homes, without checking how other segments are represented. What is the principal concern?
Compared with analyzing one matched sale pair for one feature, what can a suitably specified multiple regression do?
The base-period median sale price is $400,000 and the current-period median is $460,000. Compute the price relative defined as current ÷ base, as a decimal to three places.
The assessment-to-sale ratios are 0.8, 0.85, 0.9, 0.92, 0.95, 0.98, 1.02, 1.05, 1.08, 1.1, 1.15, 1.2. Calculate COD as [average absolute deviation from the median ÷ median]×100. For even n, average the two middle ratios. Report the COD as a percentage to two decimals.
Assessment-to-sale ratios are 0.94, 0.96, 0.98, 1, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Market data show a first quartile sale price of $405,000 and a third quartile sale price of $495,000. What is the interquartile range, to the nearest dollar?
Assessment-to-sale ratios are 0.96, 0.98, 0.99, 1.01. Compute the arithmetic mean. Report as a decimal ratio to three places.
Given first quartile $400,000, median $440,000 and third quartile $500,000, compute the trimean (Q1+2×median+Q3)/4. Report to the nearest dollar.
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