Regression Predictions: Point Estimates, Intervals, and Extrapolation

All flashcards

Regression point prediction

A single fitted estimate of the dependent variable at specified independent-variable values.

Confidence interval for a mean response

An interval estimating the average response for cases with a selected input value.

Prediction interval

An interval estimating the response of one individual case at a selected input value.

Extrapolation

Using a fitted regression equation for an input outside the observed data range.

Gauss–Markov theorem

Under its assumptions, least squares has minimum variance among linear unbiased estimators.

Quiz questions

1. In simple linear regression, what does a confidence interval at input value $x_p$ estimate?Answer: The mean response for cases with input value $x_p$.
2. Why is a prediction interval wider than a confidence interval for the mean response at the same input and confidence level?Answer: It includes the variability of individual outcomes around the mean response.
3. All else equal, what happens to regression intervals as the selected input moves farther from the sample mean input?Answer: They generally become wider.
4. A regression model was fitted using input values from 65 to 75. What is a key concern when predicting at an input of 82?Answer: The prediction is an extrapolation, so the observed relationship may not continue there.
5. In simple linear regression, which degrees of freedom are used for the $t$ critical value in the stated interval formulas?Answer: $n-2$