(e) What assumptions are usually made about the error terms of linear regression models? (6%)

(a) X and Y are independent chi-square random variables with 2 and 5 degrees of freedom, respectively. What kind of distribution the random variable X+Y will be? (6%)

(b) What is the meaning of adjusted R² for the regression analysis?(6%)

(c) Let the variables X₁, X₂, and X₃ have the variances σ₁² = 1.0, σ₂² = 3.0, σ₃² = 5.0. Let these variables be independent and Y = 13 - 2X₁ + 3X₂ - 10X₃. Find the variance of Y.(6%)

(d) Let Y = β₀ + β₁X₁ + β₂X₂ + β₃X₃ be a multiple regression model with X₃ =.What would be happened when we try to estimate the parameters of this model? and explainwhy? (6%)

2. (10%) The annual rainfall (in inches) in a certain region is normally distributed with μ = 30, σ = 3. What is the probability that in 2 of the next 4 years the rainfall will exceed 35 inches? Assume that the rainfalls in different years are independent.

3. (10%) If the density function of X equals Please find the value of c. What is probability for X>2 (i.e. P(X>2))?

(a) What is the distribution of ? (5%)

(b) If= 4, give an interval that, with 90 percent confidence, will contain the value ofXₙ₊₁. (5%)

5. (10%) A simple random sample of 500 students from a university was selected to estimate the mean living expense during last academic year. Another simple random sample of 550 students was selected, independently of the first, to estimate the mean living expense during this academic year. The sample results (expressed in thousand dollars) were as follows:Construct a 95% confidence interval for μ₂ - μ₁. Can one conclude that the mean living expense  per student has changed between the two years at α = 0.05?

(a) Estimate the components of variance, within and between groups, and the total mean. (5%)