Inference in Regression

Statistics · Axiom Academy

LESSON Inference in Regression Understanding hypothesis tests, confidence intervals, and statistical significance in linear regression The most common question in regression: Is there a real relationship between X and Y? We test this using the null hypothesis that the true slope β₁ equals zero (no relationship). If the null is true, the regression line is horizontal, and X has no predictive value for Y. To test whether β₁ = 0, we calculate a t-statistic that measures how many standard errors the estimated slope b₁ is away from zero. where b₁ is the estimated slope and SE(b₁) is its standard error. A large |t| value (far from 0) provides evidence against H₀. We compare the t-statistic to a t-distribution with n-2 degrees of freedom. 3. Confidence Intervals for Slope Instead of just testing whether β₁ = 0, a confidence interval gives us a range of plausible values for the true slope parameter. where t* is the critical value from t-distribution with n-2 df (≈ 1.96 for large n) This interval tells us: "We are 95% confident that the true slope lies within this range." 4. Interpreting Significant Slopes When we find a statistically significant slope (p < 0.05), what does it mean? The observed relationship is unlikely due to chance alone We have evidence that X and Y are related in the population But it doesn't tell us if the effect is practically important Example: If b₁ = 0.85 with p = 0.003 for predicting test scores from study hours:

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