Final answer:
The (X'X) matrix with diagonal entries of 12, 8, 8, and 8 represents data collected from 36 data points. The multiple linear regression model has 4 parameters. The variance of the intercept can be calculated using the formula Var(b0) = s^2 * ((1/n) + (X'X)^(-1)(1/n)).
Step-by-step explanation:
In a regression situation, the (X'X) matrix is diagonal with the following entries on the diagonal: 12, 8, 8 and 8. The diagonal elements of (X'X) represents the sum of the squared values of the corresponding predictor variables. Since the diagonal entries are 12, 8, 8, and 8, it means that you have collected data from 12 + 8 + 8 + 8 = 36 data points.
In multiple linear regression, the number of parameters is equal to the number of predictor variables plus one for the intercept term. In this case, since you have three predictor variables (X1, X2, X3), the number of parameters in your multiple linear regression model is 3 + 1 = 4.
The variance of the intercept can be calculated using the formula Var(b0) = s^2 * ((1/n) + (X'X)^(-1)(1/n)), where s^2 is the residual variance and n is the number of data points. Since the (X'X) matrix is diagonal with the diagonal entries 12, 8, 8, and 8, the inverse of (X'X) is also a diagonal matrix with the diagonal entries (1/12), (1/8), (1/8), and (1/8). Plugging in the values, the variance of the intercept is Var(b0) = s^2 * ((1/n) + (1/12 + 1/8 + 1/8 + 1/8)*(1/n)).