CVEN9407 Chap.6 Inference and Prediction from a Fitted Regression
Inference and Prediction from a Fitted Regression
A fitted line becomes evidence only when its uncertainty is measured. This chapter explains why the errors are assumed normal and independent, how that assumption turns estimates into t and F statistics, and why maximum likelihood understates the error variance in small samples.
Continuing the trip-generation data, it tests whether the slope differs from zero and whether it lies below one, builds its confidence interval, and confirms the F ratio. It then separates prediction of a mean response from prediction for one household and shows how to report a regression.
This is the chapter that turns estimates into statements a reader can act on.
In the first project report, each coefficient needs a standard error and a test, the model needs an overall F test, and any forecast for your area needs an interval. The same t and F logic carries into the multiple regression and diagnostics of the next chapter without change.
What this chapter covers
- 01
The classical normal linear regression model
- 02
Estimating the error variance and standard errors
- 03
Slope t tests and confidence intervals
- 04
The F ratio and analysis of variance
- 05
Mean and individual prediction
- 06
Reporting a regression
Worked example · free
Testing and bounding a slope from 22 observations
- 1t = 0.48 / 0.16 = 3.00, with n - 2 = 20 degrees of freedom.
- 1Since 3.00 > 2.086, reject a zero slope at the 5% level.
- 1Interval: 0.48 +/- 2.086 x 0.16 = 0.48 +/- 0.334, which is 0.146 to 0.814 extra crashes a year per thousand vehicles a day.
Key terms
- Standard error
- A standard error is the estimated standard deviation of an estimator's sampling distribution, used to scale t statistics.
- Confidence band
- A confidence band is the set of interval limits around a fitted line, widening with distance from the mean of X.
- Mean prediction
- Mean prediction estimates the average value of Y for all units at a chosen value of X.
- Individual prediction
- Individual prediction estimates the value of Y for a single new unit and carries the extra variance of one disturbance.
- Analysis of variance
- Analysis of variance splits the total variation in the dependent variable into the part the regression explains and the residual part.
Inference and Prediction from a Fitted Regression FAQ
Why is the error variance divided by n minus two?
Two coefficients, the intercept and the slope, are estimated from the same data before the residuals are computed, which uses up two degrees of freedom. Dividing by n minus two makes the estimate of the error variance unbiased.
Why is the interval for one new case wider than the interval for the mean?
The mean prediction is uncertain only because the line is estimated. A single new case adds its own random disturbance on top of that, so its interval must cover both sources of variation.
Are the t test and the F test different with one regressor?
No. With a single explanatory variable the F statistic equals the square of the t statistic on the slope, so the two give the same decision; F becomes useful for joint tests once there are several regressors.
What should a regression results table in my report include?
Each coefficient with its standard error and t ratio or p value, the number of observations, R squared and adjusted R squared, the F statistic, and a note defining the variables and significance markers. The brief asks for clean tables, not pasted software output.
Can I forecast outside the range of my data?
You can compute a number, but the prediction interval widens quickly away from the mean of X, and the straight-line form has never been tested there. Flag such forecasts clearly in a report and prefer statements within the observed range.
Assessment move
Practise turning raw regression output into three sentences: the effect with its interval, the test result with its p value, and the fit. Rework the prediction formulas at a value of X near the mean and at one far from it, and notice how the interval grows. Use that contrast when you discuss forecasts in your report. Build one results table in the format your report will use and reuse it for every regression you present.
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