Answer First
A t‑test compares an estimated coefficient to its hypothesized value, scaled by its standard error. The p‑value tells you how likely it is to observe a coefficient this extreme if the null hypothesis were true. If the p‑value is small, you reject the null.
Problem Setup
Simple regression: \[ y = \beta_0 + \beta_1 x + u. \] Hypothesis test: \[ H_0: \beta_1 = 0, \] \[ H_1: \beta_1 \neq 0. \] t‑statistic: \[ t = \frac{\hat{\beta}_1 – 0}{SE(\hat{\beta}_1)}. \] Decision rule:
- Large |t| → reject \(H_0\)
- Small p‑value → reject \(H_0\)
Step-by-Step Explanation
1. Estimate the regression
Obtain \(\hat{\beta}_1\) and its standard error \(SE(\hat{\beta}_1)\).
2. Compute the t‑statistic
Divide the coefficient by its standard error. Example: If \(\hat{\beta}_1 = 2.0\) and \(SE = 0.5\), then \(t = 4.0\).
3. Find the p‑value
Use the t‑distribution with \(n – 2\) degrees of freedom. Large |t| → small p‑value.
4. Compare to significance level
Common choices: 10%, 5%, 1%. If p < α → reject the null.
5. Interpret the result
Rejecting the null means the coefficient is statistically different from zero. Failing to reject means there is not enough evidence to conclude an effect.
Intuition
The t‑test asks: “Is this coefficient big relative to its noise?” If the coefficient is large compared to its standard error, it is unlikely to be zero.
Common Exam Mistakes
- Thinking a large coefficient automatically means significance.
- Confusing p‑values with effect size.
- Using the wrong degrees of freedom.
- Interpreting ‘fail to reject’ as ‘accept the null.’
Final Summary
To compute a t‑test, divide the coefficient by its standard error. The p‑value measures how extreme the t‑statistic is under the null. Small p‑values indicate statistical significance.
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