Track 9 · Correlation and Regression · Lesson 8 of 15

Inference for regression

Test and estimate a regression slope with its standard error, and choose between a confidence interval for a mean response and a prediction interval for a new observation.

35 minAdvanced8 graded exercisesMembers

A brand manager wants to know two different things from last year's data: how precisely the sales gain per thousand SAR of TV spend is known, and what range of sales to plan for in a week with 60 thousand SAR of TV advertising. The first is a question about a slope; the second is about one future week — and the second interval is always much wider.

The fitted slope $b_1$ is an estimate of the population slope $\beta_1$ in the model $y = \beta_0 + \beta_1 x + \varepsilon$ . A different sample of weeks would give a different line, so $b_1$ has a sampling distribution. Under the LINE assumptions (see Residual analysis) its standard error, test and interval follow the familiar $t$ pattern with $n - 2$ degrees of freedom.

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It has 8 graded exercises with feedback, hints and worked solutions. After it you will be able to:

Compute the standard error, t statistic and confidence interval of a slope
Explain why a spread-out predictor gives a more precise slope
Choose between a confidence interval for a mean response and a prediction interval for one observation
Explain why prediction intervals stay wide with more data and fail outside the data

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Formula sheet

Standard error, test of H0: β₁ = 0, and confidence interval for the slope
$SE(b_1) = \frac{s}{\sqrt{S_{xx}}}, \qquad t = \frac{b_1 - 0}{SE(b_1)}, \qquad b_1 \pm t^{*}_{n-2}\, SE(b_1)$
Confidence interval for the mean response (left) and prediction interval for a new observation (right)
$\hat{y}_0 \pm t^{*}_{n-2}\, s\sqrt{\frac{1}{n} + \frac{(x_0 - \bar{x})^2}{S_{xx}}} \qquad\text{vs}\qquad \hat{y}_0 \pm t^{*}_{n-2}\, s\sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{S_{xx}}}$

Reference

Inference for regression coefficients
Each coefficient has a standard error, a t-test of β = 0 and a confidence interval on n − p degrees of freedom.
Confidence vs prediction intervals in regression
A confidence interval estimates the mean response at x₀; a prediction interval covers a single new observation and is always wider.

Working method

Read the question twice: what is being asked, about which group or variable?
Name the method before calculating.
Keep full precision; round only the final answer.
Ask whether the answer is plausible in context.