Questions: Hypothesis Tests and Inference in Regression

Question 1

A researcher fits a simple linear regression and obtains β̂₁ = 0.45 with SE(β̂₁) = 0.18 and n = 30. The p-value for H₀: β₁ = 0 is 0.018. What is the correct interpretation of this p-value?

• A. There is a 1.8% probability that β₁ = 0 in the population
• B. If β₁ = 0, the probability of observing a t-statistic at least as extreme as the one observed is 1.8%
• C. The regression explains 98.2% of the variation in Y
• D. There is a 98.2% probability that the true slope is positive

Answer: (B)

A p-value is the probability of observing data at least as extreme as what was observed, assuming the null hypothesis is true. It is not the probability that the null hypothesis is true (option A), not a measure of model fit (option C), and not a posterior probability about the parameter (option D). Under H₀: β₁ = 0, a t-statistic this large would arise only 1.8% of the time by chance, so we reject H₀ at the 5% level.

Question 2

A confidence interval for the mean response E[Y|X=x*] and a prediction interval for a new observation at the same x* are computed. Which statement correctly describes their relationship?

• A. They are the same width, since both use the same regression equation
• B. The confidence interval is wider, because it must account for uncertainty in both β̂₀ and β̂₁
• C. The prediction interval is wider, because it must also account for the irreducible scatter of individual observations around the true line
• D. The confidence interval is wider near X̄ but the prediction interval is wider far from X̄

Answer: (C)

Prediction intervals are always wider than confidence intervals for the mean at the same X value. A confidence interval captures only parameter uncertainty — how precisely we've estimated the true mean response at x*. A prediction interval must also account for residual variation: even if we knew the true line exactly, individual observations scatter around it with variance σ². This additional source of uncertainty makes prediction intervals irreducibly wider.

Question 3

A 95% confidence interval for β₁ that does not include zero is equivalent to rejecting H₀: β₁ = 0 at the 5% significance level.

• T. True
• F. False

Answer: True

A confidence interval at level (1−α) and a hypothesis test at level α are two perspectives on the same question. The confidence interval contains exactly those values of β₁ that would not be rejected by a two-sided test at level α. If zero is not in the interval, then zero would be rejected — which is identical to rejecting H₀: β₁ = 0 at level α. Both approaches yield the same conclusion about statistical significance.

Question 4

Both confidence intervals for the mean response and prediction intervals for new observations are narrowest at the extreme ends of the observed X range.

• T. True
• F. False

Answer: False

Both intervals are narrowest at X = X̄, the mean of the predictor values, and widen as X moves away from X̄ in either direction. The OLS regression line is pinned by the data's centroid — the estimate of the mean response is most precise there. As X moves toward the extremes or into extrapolation territory, uncertainty in the slope estimate compounds, widening both intervals. The extremes of the X range are where the intervals are widest, not narrowest.

Question 5

Why does the t-test for the regression slope β₁ use n − 2 degrees of freedom rather than n − 1?

Model answer: Two parameters are estimated from the data — the intercept β̂₀ and the slope β̂₁ — and each estimated parameter costs one degree of freedom. Starting with n observations, we lose 2 degrees of freedom for the two estimated parameters, leaving n − 2 for the residual variance estimate s². The t-statistic uses s in its denominator, so it follows a t-distribution with n − 2 degrees of freedom under H₀.

Degrees of freedom count the number of independent pieces of information remaining after estimating parameters. A one-sample t-test uses n − 1 df because only the mean is estimated. Simple linear regression estimates two parameters (intercept and slope), so n − 2 df remain. Multiple regression with p predictors (plus intercept) uses n − p − 1 df for the same reason. This matters for the critical value and p-value of the test.