Under the usual regression assumptions, at the same predictor settings and nominal coverage, how does an interval for one future response differ from a confidence interval for the mean response?
Correct Answer
C) A prediction interval covers a single new observation
Under the same fitted regression, predictor settings, nominal coverage and usual error assumptions, an interval for one future response includes both uncertainty in the fitted mean and individual-response variation. It is wider than the corresponding confidence interval for the mean response. It is not an interval for the sample mean, and neither construction dispenses with uncertainty estimation. Comparing intervals at different coverage levels or settings would not support the same width claim.
Why This Is the Correct Answer
Option C is correct because a prediction interval covers a single new observation, while a confidence interval covers the mean response at the given predictor settings. Because the prediction interval adds individual-response variation to the uncertainty in the fitted mean, it is wider than the corresponding confidence interval.
Why the Other Options Are Wrong
Option A: A prediction interval is the narrower of the two
Option A, that a prediction interval is the narrower of the two, reverses the relationship. At the same settings and coverage, the prediction interval is wider because it includes individual-response variation as well as uncertainty in the fitted mean.
Option B: A prediction interval applies to the sample mean
Option B, that a prediction interval applies to the sample mean, is wrong. A prediction interval targets one future response, not the sample mean or the mean response.
Option D: A prediction interval requires no standard error at all
Option D, that a prediction interval requires no standard error, is wrong. A prediction interval is built from a standard error that combines uncertainty in the fitted mean with the residual variation.
One house wobbles more
Predicting one new sale carries the uncertainty of the average plus that sale's own scatter, so its interval is wider. Predicting the average carries only the first part.
How to use: Use this to remember which interval is wider and what each covers.
Exam Tip
When asked about a single future property, think prediction interval; when asked about the average for properties like it, think confidence interval.
Common Mistakes to Avoid
- -Believing the prediction interval is narrower
- -Confusing the mean response with a single future response
- -Thinking a prediction interval needs no standard error
Concept Deep Dive
Analysis
This question tests the difference between a prediction interval for a single future observation and a confidence interval for the mean response in regression. A prediction interval includes both the uncertainty in the fitted mean and the variation of individual responses around it, so it is wider at the same settings and coverage.
Background Knowledge
In regression, a confidence interval for the mean response describes uncertainty about the average outcome at given predictor values. A prediction interval describes where a single new observation is likely to fall and must also account for individual scatter around the regression line. Comparisons of width hold only at the same predictor settings and nominal coverage.
Real-World Application
An appraiser uses a regression to estimate the price of one specific subject home. The prediction interval for that home is wider than the confidence interval for the average price of homes with the same features, and the appraiser explains this when describing the reliability of the estimate.
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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The assessment-to-sale ratios are 0.9, 0.95, 1, 1.05, 1.15. 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.