Inference for Regression Math Example 1

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Example 1

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A regression output shows: slope b=2.5b=2.5, SEb=0.8SE_b=0.8, n=30n=30. Test H0:β=0H_0: \beta=0 vs Ha:β≠0H_a: \beta \neq 0 at α=0.05\alpha=0.05 using a t-test.

Solution

  1. 1
    Test statistic: t=b−β0SEb=2.5−00.8=3.125t = \frac{b - \beta_0}{SE_b} = \frac{2.5 - 0}{0.8} = 3.125
  2. 2
    Degrees of freedom: df=n−2=30−2=28df = n - 2 = 30 - 2 = 28
  3. 3
    Critical value: t0.025,28∗≈2.048t^*_{0.025, 28} \approx 2.048 (two-tailed at α=0.05\alpha=0.05)
  4. 4
    Since ∣t∣=3.125>2.048|t| = 3.125 > 2.048, reject H0H_0; the slope is significantly different from zero

Answer

t=3.125>2.048t=3.125 > 2.048. Reject H0H_0. The slope is statistically significant at α=0.05\alpha=0.05.
Testing whether the slope equals zero tests whether x is a useful predictor of y. Rejecting H0:β=0H_0: \beta=0 means the linear relationship exists (in the population). df = n-2 because two parameters (slope and intercept) are estimated.

About Inference for Regression

Using hypothesis tests and confidence intervals to draw conclusions about the true population slope β1\beta_1 of the linear relationship y=β0+β1x+εy = \beta_0 + \beta_1 x + \varepsilon, based on sample data.

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More Inference for Regression Examples

Example 2 hard

Construct a 95% confidence interval for the slope [formula] given: [formula], [formula], [formula],

Example 3 easy

List the four conditions for valid regression inference and explain why each must be checked.

Example 4 hard

A regression of salary on years of experience gives: [formula], [formula], slope p-value=0.001. A co

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