Interpretation of the Slope Coefficient:

The slope coefficient for $X$ (income) is 20.42. This indicates that for every unit increase in income (e.g., 1 dollar), the loan amount increases by 20.42 units on average, assuming all other factors remain constant.

Statistical Significance of the Slope Coefficient:

• The t-statistic for the slope is 31.58, and its associated p-value is $3.405 \times 10^{-156}$, which is far smaller than any conventional significance level (e.g., 0.05, 0.01, or 0.001).
• This extremely small p-value suggests that the relationship between income and loan amount is highly statistically significant. We reject the null hypothesis that the slope coefficient is zero, confirming a strong positive association.

Interpretation of $R^2$ (Coefficient of Determination):

• The $R^2$ value is 0.476, which means approximately 47.6% of the variance in the loan amount ($Y$) is explained by the variance in income ($X$) based on the model.
• While this is a substantial proportion, it also implies that 52.4% of the variance in loan amounts is due to other factors not included in the model.

P-Level (Significance of the Model):

• The Significance F for the overall model is $3.405 \times 10^{-156}$, indicating that the regression model as a whole is statistically significant. The probability of observing such a strong relationship by random chance is essentially zero.

Final Insights:

1. The positive slope demonstrates that income is a strong predictor of loan amount.
2. The extremely small p-values for both the slope coefficient and the overall model confirm their statistical significance.
3. While $R^2 = 0.476$ suggests the model is useful, there is still room for improvement by including additional predictors to better explain variations in loan amounts.

Do you have any questions or want to explore further details? Here are some related questions:

1. What does Adjusted $R^2$ tell us, and why is it slightly lower than $R^2$?
2. How would you assess multicollinearity if there were more predictors in the model?
3. How might you improve the model to explain a greater proportion of the variance in loan amounts?
4. Can we interpret the intercept meaningfully in this context? Why or why not?
5. How does the Standard Error of the regression inform us about prediction accuracy?

Tip: Always look at residual plots to assess whether the linear regression assumptions (e.g., homoscedasticity, normality) hold true for your model!