Question

If $\hat{b}_1 > 2\text{S.E}(\hat{b}_1)$ then we}

• A. reject $H_0$
• B. reject $H_A$
• C. accept $H_0$
• D. accept $H_A$

Hint:

The "rule of thumb" in regression analysis states that if an estimated coefficient is more than twice its standard error, it is statistically significant at the $5\%$ level, allowing us to reject the null hypothesis.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In linear regression, $\hat{b}_1$ represents the estimated regression coefficient, and $\text{S.E}(\hat{b}_1)$ is its standard error.
To test whether the independent variable has a significant effect on the dependent variable, we set up hypotheses:
- Null Hypothesis ($H_0$): $b_1 = 0$ (no relationship).
- Alternative Hypothesis ($H_A$): $b_1 \neq 0$ (significant relationship).

Step 2: Detailed Explanation:
The test statistic (t-statistic) is calculated using the formula:
\[ t = \frac{\hat{b}_1}{\text{S.E}(\hat{b}_1)} \]
The condition given is:
\[ \hat{b}_1 > 2\text{S.E}(\hat{b}_1) \implies \frac{\hat{b}_1}{\text{S.E}(\hat{b}_1)} > 2 \implies t > 2 \]
In statistics, for a two-tailed test at a standard $5\%$ significance level, the critical value of $t$ is approximately $1.96$ (often rounded to $2$).
Since the calculated $t$-value ($t > 2$) exceeds the critical value, the result is statistically significant.
Therefore, we reject the null hypothesis ($H_0$).

Step 3: Final Answer:
We reject $H_0$, which corresponds to Option (A).