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Significance Tests for the Slope

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Testing for a linear relationship

The usual test asks whether there is a linear relationship at all: H₀: β = 0 (no linear relationship) versus Hₐ: β ≠ 0 (a linear relationship exists). Rejecting H₀ means the data provide convincing evidence that the slope is not zero — that x and y are linearly related in the population.

The t statistic and computer output

The t statistic is b divided by its standard error: t = b / SE_b (using β₀ = 0). Regression output prints this t and a two-sided p-value in the explanatory variable’s row. To convert to a one-sided test, halve the reported p-value (when the sample slope is in the direction of Hₐ). Compare the p-value to α as usual.

t test for the slope
t = b / SE_b · df = n − 2 · (printed p-value is two-sided)
For H₀: β = 0. The standard computer output p-value tests β ≠ 0; for a one-sided alternative, divide it by 2 if the slope goes the predicted way.
Worked example

A regression of exam score on hours studied (n = 15) gives b = 5.0, SE_b = 2.0. Test H₀: β = 0 against Hₐ: β ≠ 0 at α = 0.05.

  1. 1.Hypotheses: H₀: β = 0 (no linear relationship) vs. Hₐ: β ≠ 0. Assume LINER conditions are met.
  2. 2.t = b/SE_b = 5.0/2.0 = 2.5, with df = n − 2 = 13.
  3. 3.Two-sided p-value = 2·P(t₁₃ > 2.5) ≈ 2(0.013) ≈ 0.026.
  4. 4.Since 0.026 < 0.05, reject H₀.
Answer: t = 2.5 with df = 13 gives a two-sided p-value ≈ 0.026 < 0.05, so we reject H₀. There is convincing evidence of a (positive) linear relationship between hours studied and exam score.
Checkpoint

For the standard test of a regression slope, what are the null and alternative hypotheses?

Watch out

The p-value printed in regression output is two-sided (for Hₐ: β ≠ 0). If your alternative is one-sided (β > 0 or β < 0) and the sample slope matches that direction, divide the printed p-value by 2. Reporting the two-sided value for a one-sided test is a common slip.

Checkpoint

Regression output gives slope b = 6.0 with standard error SE_b = 1.5. What is the t statistic for testing H₀: β = 0?

Answer the 2 checkpoints as you read.

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