The t-statistic is a test statistic with a t-distribution. In the context of the linear model it is used to test whether a b-value is significantly different from zero; which of the following best complements this statement?

The t-statistic is a test statistic with a t-distribution. In the context of the linear model it is used to test whether a b-value is significantly different from zero; which of the following best complements this statement?

• A. It tests whether the intercept differs from zero
• B. It tests whether the slope is zero
• C. It tests whether the difference between two means is significantly different from zero
• D. It tests whether the variance of residuals is zero

Answer: (C)

In a linear model, each regression coefficient (the b value) is tested with a t-statistic to see if it differs from zero. That means the predictor has a real association with the outcome. When that predictor is a binary indicator of group membership, the coefficient for that predictor essentially captures the difference in the outcome between the two groups, holding other variables constant. So testing whether the coefficient is different from zero is exactly testing whether the two group means differ significantly. This is why the statement about testing a b value complements by linking the coefficient test to a difference-of-means question. The other options shift the focus away from the two-group contrast or from what the t statistic is testing: testing an intercept differs from zero, or testing the slope for a continuous predictor is a related concept but doesn’t specifically articulate the two-group mean difference; and testing whether the residual variance is zero is not what the t statistic assesses.

In a linear model, each regression coefficient (the b value) is tested with a t-statistic to see if it differs from zero. That means the predictor has a real association with the outcome. When that predictor is a binary indicator of group membership, the coefficient for that predictor essentially captures the difference in the outcome between the two groups, holding other variables constant. So testing whether the coefficient is different from zero is exactly testing whether the two group means differ significantly. This is why the statement about testing a b value complements by linking the coefficient test to a difference-of-means question.

The other options shift the focus away from the two-group contrast or from what the t statistic is testing: testing an intercept differs from zero, or testing the slope for a continuous predictor is a related concept but doesn’t specifically articulate the two-group mean difference; and testing whether the residual variance is zero is not what the t statistic assesses.