True or false: We can use a t-test to test whether the regression coefficient beta = 0.
The correct answer and explanation is:
Correct Answer: True
Explanation (300 Words):
In the context of linear regression, we can use a t-test to determine whether a regression coefficient (β) is significantly different from zero. This is a fundamental part of regression analysis and hypothesis testing.
When performing simple or multiple linear regression, the model typically looks like: y=β0+β1×1+β2×2+⋯+βkxk+ϵy = \beta_0 + \beta_1x_1 + \beta_2x_2 + \dots + \beta_kx_k + \epsilon
Each β\beta coefficient represents the effect of an independent variable xx on the dependent variable yy, while controlling for other variables in the model.
To test whether a specific coefficient βi=0\beta_i = 0, we perform a t-test with the following hypotheses:
- Null Hypothesis (H₀): βi=0\beta_i = 0 (The predictor has no effect)
- Alternative Hypothesis (H₁): βi≠0\beta_i \neq 0 (The predictor has an effect)
The test statistic is calculated as: t=β^iSE(β^i)t = \frac{\hat{\beta}_i}{SE(\hat{\beta}_i)}
Where:
- β^i\hat{\beta}_i is the estimated coefficient from the regression,
- SE(β^i)SE(\hat{\beta}_i) is the standard error of the estimated coefficient.
This test statistic follows a t-distribution with n−k−1n – k – 1 degrees of freedom, where:
- nn is the number of observations,
- kk is the number of independent variables.
If the computed t value is large in magnitude (greater than the critical value from the t table), or if the p-value is less than the chosen significance level (e.g., 0.05), we reject the null hypothesis and conclude that the coefficient is statistically significant — i.e., there is evidence that the predictor variable is associated with changes in the outcome.
Therefore, it is true that a t-test can be used to test whether β=0\beta = 0 in regression analysis.