Question

Use the computer output (from different computer packages) to estimate the intercept \(\beta_{0},\) the slope \(\beta_{1},\) and to give the equation for the least squares line for the sample. Assume the response variable is \(Y\) in each case. $$ \begin{aligned} &\text { The regression equation is }\\\ &Y=808-3.66 \mathrm{~A}\\\ &\begin{array}{lrrrr} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 807.79 & 87.78 & 9.20 & 0.000 \\ \text { A } & -3.659 & 1.199 & -3.05 & 0.006 \end{array} \end{aligned} $$

Short answer

The intercept \(\beta_{0}\) in the equation is 808, the slope \(\beta_{1}\) is -3.66, and the equation for the least squares line is \(Y = 808 - 3.66A\).

Step-by-step solution

Identify the Intercept and Slope

Look at the regression equation. The value that is not associated with a variable (in this case, 808) is the intercept \(\beta_{0}\). It aligns with the 'Constant' row in the table, further confirming this. The value that is linked to a variable (in this case, -3.66 with A) is the slope \(\beta_{1}\). This aligns with the 'A' row in the table.

Interpret the Intercept and Slope

The intercept (\(\beta_{0}\) = 808) represents the predicted value of \(Y\) when \(A\) is equal to 0. The slope (\(\beta_{1}\) = -3.66) means that for each one-unit increase of \(A\), \(Y\) will decrease by approximately 3.66, assuming all other predictors remain constant.

Form the Least Squares Line

The equation for the least squares line was given from the start: \(Y = 808-3.66 \times A\). This means that for each unit \(A\) increases, the predicted value of \(Y\) decreases by 3.66. And if \(A\) is 0, the predicted value of \(Y\) is 808, hence acting as the starting point (intercept).

Key concepts

Regression Analysis

Regression analysis is a form of predictive statistical technique that examines the relationship between a dependent variable and one or more independent variables. The goal is to model this relationship in a way that we can use the model to predict values of the dependent variable based on known values of the independent variables. The least squares regression method is particularly popular, as it determines the line that minimizes the sum of the squares of the residuals—the differences between the observed values and the values predicted by the model.

Let's delve into our example. When we are given output from statistical software, it usually includes the equation for a line of best fit. This line represents the regression line generated by the least squares method. In our case, the equation is given as Y = 808 - 3.66 * A. This is the line that best approximates the relationship between variable A and the response variable Y in our sample data. This equation will allow us to predict Y for any given value of A, within the context of the data.

Intercept

In regression analysis, the intercept (often denoted as \( \beta_{0} \) in equations) is the point at which the regression line crosses the y-axis. It represents the expected value of the dependent variable when all the independent variables are equal to zero. In our exercise, the intercept is estimated to be 808.

This figure can tell us a lot about the relationship we are studying. For instance, if we are looking at the relationship between the number of hours studied (\( A \)) and test scores (\( Y \)), an intercept of 808 might suggest the expected test score if no hours were invested in studying whatsoever. However, caution should be exercised with interpretation, especially if an intercept doesn't make practical sense given the context, or if the values of the independent variables never actually equal zero within the realm of the data.

Slope

The slope of the regression line represents the change in the dependent variable for a one-unit change in the independent variable. It's often denoted as \( \beta_{1} \) when referring to the first independent variable in a model. In our example, the slope is -3.66, which is associated with the variable \( A \).

What does this number tell us? A slope of -3.66 implies that for each additional unit increase in \( A \), the variable \( Y \) is expected to decrease by 3.66 units, assuming all other variables in the model are held constant. This relationship is crucial for understanding how changes in one variable are associated with changes in another. If \( A \) represents hours studied, and \( Y \) is test scores, then a negative slope might suggest that beyond a certain point, more hours lead to fatigue and lower scores—though we have to be cautious about drawing such conclusions without further analysis.

Statistical Hypothesis Testing

Statistical hypothesis testing in the context of regression allows us to evaluate the significance of our regression coefficients - essentially testing whether there is enough evidence to suggest a relationship between the dependent and independent variables in the population from which our sample was drawn. In our output, we have a 'P' value associated with each coefficient — this P value tells us the probability of observing the data (or something more extreme) assuming the null hypothesis is true. The null hypothesis here would typically be that the coefficient (intercept or slope) is equal to zero, suggesting no effect.

In our example, the P values next to the intercept and slope are both very small (0.000 for intercept and 0.006 for slope) which implies that the chances of seeing such results due to random chance alone are low. Consequently, we have evidence to reject the null hypothesis and conclude that the coefficients are significantly different from zero, indicating a statistically significant relationship between \( A \) and \( Y \) in our regression model.