# Midterm Study GuideFeb 01 2019

The midterm is closed book.
I will provide statistical tables if you need them (so you should know how to use them).
You should bring a calculator (although arithmetic errors are generally forgiven).

Things you will not have to do:

• write any R code
• invert more than a 2x2 matrix
• perform matrix arithmetic on more than a 3x3 matrix

You should be able to:

• State the multiple regression model in matrix form along with the assumptions on the errors and design matrix.

• Describe the entries in the design matrix given a model and study description.

• Derive the least squares estimates.

• Define fitted values and residuals.

• Describe the difference between random errors and residuals.

• Derive the mean and variance-covariance matrix of the least squares estimates in multiple linear regression.

• State the Gauss-Markov theorem and discuss it’s consequences in practice.

• State the form and properties of the estimate for the variance of the errors.

• Describe why using lm() in R is preferable to performing the matrix algebra $$\left(X^TX\right)^{-1}X^TY$$.

• State the distribution of the least squares estimates under the assumption of Normal errors.

• Identify properties of the least squares estimates (i.e. form of the estimates, mean, variance and distribution of the estiamtes, unbiasedness, BLUE, etc.) that rely on the Normality assumption.

• Describe the consequences of having orthogonal columns in the design matrix.

• State the null distribution of t-statistics and F-statistics in hypothesis tests relevant to multiple linear regression models.

• Construct t-based confidence intervals and hypothesis tests on individual parameters, or linear combinations of individual parameters, given either R output, or the neccessary estimates, and $$(X^TX)^{-1}$$

• Construct prediction intervals for the mean response or a future response, given either R ouput, or the neccessary estimates, and $$(X^TX)^{-1}$$

• Discuss the difference between a interval for the mean response and an interval for a future response.

• Discuss ways in which a prediction model can go wrong.

• Interpret a confidence interval or prediction interval in the context of a study.

• Comment on the conclusion a hypothesis test would reach based on the result of a confidence interval/region.

• Conduct an F-test to compare two models.

• Interpret the result of an F-test in context of a study.

• State the null and alternative hypotheses in the overall regression sum of squares F-test.

• Find the linear parameteric function test equivalent to a model test and vice versa (i.e. HW#4 Q1).