To determine whether we reject or fail to reject $H_0$ (that the slope coefficient is $3.94$) at the 5% significance level, we need to perform a hypothesis test for the slope coefficient $\beta_1$. The test statistic for this hypothesis test is given by:

$t = \frac{\hat{\beta}_1 - \beta_1}{SE(\hat{\beta}_1)}$

where:

• $\hat{\beta}_1 = 10.21$ (the estimated slope coefficient),
• $\beta_1 = 3.94$ (the hypothesized slope coefficient under $H_0$),
• $SE(\hat{\beta}_1) = 43.41$ (the standard error of the slope, given in parentheses).

Step 1: Calculate the t-Statistic

Using the provided values: $t = \frac{10.21 - 3.94}{43.41} = \frac{6.27}{43.41} \approx 0.1444$

Step 2: Compare with the Critical Value

For a two-tailed test at the 5% significance level and $N - 2 = 40 - 2 = 38$ degrees of freedom, we can look up the critical $t$-value for a 5% significance level (or use a table to approximate). For 38 degrees of freedom, the critical value for $t$ is approximately $\pm 2.024$.

Since $|t| = 0.1444$ is far less than $2.024$, we do not reject $H_0$.

Conclusion

The correct answer is: (b) do not reject $H_0$.

Would you like further details on hypothesis testing or interpreting regression outputs?

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1. How would changing the significance level impact the decision?
2. How does the R-squared value affect the interpretation of this model?
3. Why is the standard error important in hypothesis testing?
4. What other factors could impact the reliability of this regression analysis?
5. How would a larger sample size affect the accuracy of our estimate?

Tip: In hypothesis testing, always compare the calculated t-statistic to the critical value based on your chosen significance level.