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The Sampling Distribution of the Slope

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From sample line to population line

Behind a scatterplot is a true linear relationship with a population slope β (a parameter). Each sample produces a least-squares slope b (a statistic) that estimates β. Because b varies from sample to sample, it has a sampling distribution — and inference for regression is really inference about β using b, exactly parallel to using x-bar for μ.

The sampling distribution of b

When the regression conditions hold, the sampling distribution of the slope b is centered at β (b is unbiased) and has a standard deviation estimated by the standard error of the slope, SE_b. Standardizing b uses a t-distribution with n − 2 degrees of freedom — we subtract 2 because both a slope and an intercept were estimated from the data.

Standardized slope and df
t = (b − β₀) / SE_b · df = n − 2
b is the sample slope, β₀ the hypothesized slope (usually 0), and SE_b the standard error of the slope (read from computer output). Two parameters estimated → df = n − 2.
Worked example

Computer output for a regression of y on x (n = 20) reports slope b = 2.5 with SE_b = 0.5. Identify the parameter being estimated and the degrees of freedom for inference.

  1. 1.The sample slope b = 2.5 estimates the true population slope β, the parameter of interest.
  2. 2.The standard error SE_b = 0.5 describes how much b would vary from sample to sample.
  3. 3.Degrees of freedom for slope inference: df = n − 2 = 20 − 2 = 18.
Answer: b = 2.5 is the statistic estimating the population slope β; SE_b = 0.5 measures its sample-to-sample variability; and inference uses a t-distribution with df = n − 2 = 18.
Checkpoint

In regression inference, what does the population slope β represent?

Tip

Regression inference is the familiar t-procedure in disguise: b plays the role of x-bar, β the role of μ, SE_b the role of s/sqrt(n). The only new wrinkle is df = n − 2 because two parameters (slope and intercept) were estimated.

Checkpoint

A regression uses n = 25 data points. How many degrees of freedom are used for inference about the slope?

Answer the 2 checkpoints as you read.

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