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Regression coefficients represent the strength and direction of the relationship between each predictor variable and the response variable in a regression model.
In simple terms, the coefficient tells you how much the response variable changes with a one-unit increase in the predictor, keeping all other predictors constant.
For example, if a coefficient is -1.00, as it is for \(X_1\) in our exercise, it indicates that for every one-unit increase in \(X_1\), the predicted value of the response variable decreases by 1 unit.

• A positive coefficient suggests a direct relationship, meaning as the predictor increases, so does the response variable.
• A negative coefficient indicates an inverse relationship, where an increase in the predictor is associated with a decrease in the response variable.

Understanding these coefficients helps in determining the impact of each variable in the model and is critical when it comes to hypothesis testing.

The F-test is crucial in determining if the overall regression model is statistically significant.
It tests the null hypothesis that all the regression coefficients in the model are equal to zero.
Simply put, it assesses whether the independent variables together have a significant effect on the dependent variable.The F-statistic is calculated using the formula:\[ F = \frac{MSR}{MSE} \]where MSR is the Mean Square for Regression and MSE is the Mean Square for Error. In the exercise, we computed this as approximately 18.00. This value is then compared to a critical F-value obtained from F-distribution tables, based on the chosen significance level and degrees of freedom.

• If the calculated F-value is greater than the critical value, we reject the null hypothesis.
• This indicates that at least one of the predictors has a non-zero coefficient, meaning the model has predictive power.

In this exercise, since the calculated F (18.00) was greater than the critical F-value (3.16), the result was significant.

The t-test is used to determine if individual regression coefficients are significantly different from zero.
This helps identify which specific predictors significantly contribute to the model, beyond assessing the model as a whole.
For each coefficient, the t-statistic is calculated by dividing the coefficient by its standard error.The formula is:\[ t = \frac{\text{Coefficient}}{\text{StdDev}} \]In the given exercise, each predictor variable had its own t-value calculated. These values were compared against a critical t-value for the chosen significance level, obtained from t-distribution tables (±2.101 in this example):

• For \(X_1\), \(|t| = 4.00 > 2.101\), making it significant.
• \(X_2\) with \(|t| = 1.50 < 2.101\), was not significant, suggesting it might not add value to the model.
• \(X_3\) had \(|t| = 3.00 > 2.101\), hence significant.

The t-test is essential for optimizing the model by identifying and potentially removing non-contributing predictors.

The significance level, often denoted by \( \alpha \), indicates the probability of rejecting the null hypothesis incorrectly (Type I error).
In hypothesis testing, it's typically set at commonly used values like 0.05 or 0.01, representing a 5% or 1% risk.
It provides a threshold for decision-making.In our exercise, a 0.05 significance level was used:

• For the global F-test, this significance level helped decide to reject the null hypothesis, verifying that the model was useful.
• For individual t-tests, it determined which coefficients (predictors) were significant by comparing calculated t-values against the critical t-values.

Significance levels are crucial in statistical tests, as they help decide the reliability and robustness of the inferences drawn from the data. Understanding them ensures informed decisions during hypothesis testing.