Part 1: Regression Model Construction

The given regression equation is:

$y = -9.632 + 1.328x_1 + 0.722x_2 - 0.090x_3 - 0.494x_4$

This regression model is already constructed using all three independent variables. Each coefficient corresponds to an independent variable, indicating its effect on the dependent variable $y$, while the constant term is $-9.632$.

Part 2: Hypotheses for Testing Significance

To test the significance of each independent variable at $\alpha = 0.05$, the null and alternative hypotheses are as follows:

• Null Hypothesis ($H_0$): $\beta_j = 0$ (The population regression coefficient for the $j$-th variable is not significant.)
• Alternative Hypothesis ($H_1$): $\beta_j \neq 0$ (The population regression coefficient for the $j$-th variable is significant.)

Here $\beta_j$ is the population regression coefficient for the $j$-th independent variable.

Part 3: Test Statistics

The test statistics for each coefficient are given as:

Conclusion: Statistical Significance Testing

1. The critical value for a $t$-test depends on the degrees of freedom (usually $n - k - 1$, where $n$ is the sample size and $k$ is the number of independent variables). However, as a general rule at $\alpha = 0.05$, any test statistic greater than approximately $\pm 1.96$ in absolute value is significant for large samples.

2. Evaluating the significance of the coefficients:

   • $x_1$: $t = 5.45$ (significant since $|t| > 1.96$)
   • $x_2$: $t = 4.30$ (significant since $|t| > 1.96$)
   • $x_3$: $t = 0.36$ (not significant since $|t| < 1.96$)
   • $x_4$: $t = -2.05$ (significant since $|t| > 1.96$)

Final Model Interpretation

• Significant variables ($x_1, x_2, x_4$) contribute meaningfully to explaining the variance in $y$.
• $x_3$ does not contribute significantly and could potentially be removed for a simplified model, depending on further analysis.

Questions for Further Exploration

1. What is the sample size used, and how does it affect the degrees of freedom?
2. What are the confidence intervals for each coefficient?
3. How do the residuals behave for this model? Is there any sign of heteroscedasticity?
4. Can we check multicollinearity among the independent variables using variance inflation factors (VIF)?
5. How does removing $x_3$ from the model affect $R^2$ and adjusted $R^2$?

Tip

Always assess the practical significance of a regression coefficient in addition to its statistical significance. A statistically significant coefficient might have little real-world impact if the effect size is small.