STAT 513 -- EXAM 3 REVIEW SHEET 

0. Leftover Simple Linear Regression Topic    
  A. Prediction Interval for the Response of a New Observation at a particular x-value
    1. Prediction Error
    2. Interpretation of PI
    3. Comparison with CI for E(Y) at a particular x-value
00. General Topic in Regression
  A. Knowing how to derive Least-Squares Estimators from First Principles
    1. Write expression for SSE in terms of the beta-hats of the model
    2. Taking derivatives of SSE with respect to beta-hat(s)
    3. Setting derivative expression(s) equal to 0 and solving equation(s) for beta-hat(s)
  B. Review the various examples we did in class, on previous test, and in various homeworks
I. Multiple Linear Regression (MLR)
   A. General MLR model with k predictors
    1. General Linear Model in Matrix Terms
      a. Y vector
      b. X matrix
      c. beta vector
      d. epsilon vector
    2. Estimating the betas in MLR
      a. Formula for vector of estimated coefficients
      b. Fitting a quadratic regression model
      c. variances and covariances for estimated coefficients
    3. Properties of Least-squares Estimators in MLR
      a. Unbiasedness of estimated coefficients
      b. Formulas for SSE and MSE in MLR
      c. Distribution of estimated coefficients
      d. Distribution of (n-k-1)MSE/(sigma^2)
   B. Inference in MLR
    1. Tests and CIs about a0*beta_0 + a1*beta_1 + ... + ak*beta_k
      a. Figuring out the appropriate "a" vector for the situation
    2. Tests about an individual beta_i
      a. Tests marginal effect of individual explanatory variable
      b. "in the presence of" other predictors in the model
    3. Test statistic, rejection region for test of H_0: beta_i = 0
    4. Confidence Interval for an individual beta_j
    5. CI for the E(Y) at a particular X value
    6. Interpreting the CIs
    7. Prediction Interval for 'new' response value, Y*
   C. Reduced and Complete Models
    1. Behavior of SSE as independent variables are added to the model
    2. Writing Reduced model and Full model
    3. Testing whether some (or all) explanatory variables are useless
      a. Null and alternative hypotheses
      b. Cochran's Theorem
      c. Test statistic and Rejection region
II. Correlation Model
   A. Population correlation coefficient rho
    1. What does it measure?
    2. Its estimate, the sample correlation coefficient r
   B. Testing whether rho=0
    1. Relationship to t-test about beta_1 in SLR
   C. Testing whether rho = some nonzero number
    1. Fisher's z-transformation
    2. Finding a large-sample CI for rho using back-transformation
   D. Coefficient of Determination
    1. Formula for r^2 in SLR
    2. Interpretation of r^2
    3. Coefficient of multiple determination R^2 in MLR
III. Linear Model for ANOVA
   A. Basic Terms and Setup
    1. Factor and Levels
    2. One-way ANOVA Model
    3. Goal of ANOVA
    4. One-way ANOVA model equation
   B. ANOVA as a Linear Model
    1. Dummy variables
    2. Baseline level
    3. Comparing E(Y) across levels
    4. Testing whether betas all equal 0
    5. Structure of X matrix for an ANOVA model
    6. F-test
    7. Bonferroni approach to compare each pair of level means
IV. Inference for Categorical Data and Contingency Tables
   A. Test about Multinomial Probabilities
    1. Conditions for a Multinomial Experiment
    2. Finding expected counts under H0
    3. H_0 and H_a for Test about Multinomial Probabilities
    4. Test statistic for Test about Multinomial Probabilities
    5. Rejection region (including correct d.f.) for this test
    6. Large-sample rule of thumb
      a. Simple remedy if large-sample rule is not satisfied
   B. Chi-square Goodness of Fit Test
    1. Testing whether a particular distribution is a good model for a data set
    2. Finding probabilities under the model, and calculating expected counts under the model
    3. Rejection region (including correct d.f.) for this test
    4. Situation when parameter(s) of the model must be estimated
   C. Chi-square Test for Independence (r by c table)
    1. Formula for Cell Probabilities under Independence
    2. Calculating Observed and expected counts
    3. Test statistic and its null distribution and rejection region
    4. Large-sample checks
   E. Chi-square Test for Homogeneity (r by c table)
    1. Difference in sampling process compared to test for independence
    2. Difference in hypotheses compared to test for independence
    3. Test statistic and its null distribution and rejection region
    4. Large-sample checks