STAT 704 -- TEST 1 REVIEW SHEET

I.   Random Variables and Important Distributions
   A. Random Variables
    1. Density Function of a r.v.
    2. Expected Value of a r.v. & Properties
    3. Variance of a r.v. & Properties
    4. Standard Deviation of a r.v.
    5. Covariance and Correlation
    6. Independence (and how it's related to covariance)
   B. Linear Combinations of Random Variables
    1. Expected Value of Linear Combination
    2. Variance of Linear Combination
    3. Variance of Linear Combination of Independent r.v.'s
   C. Sample Mean Y-bar
    1. Expected value and Variance of Y-bar
    2. Central limit Theorem
   D. Normal Distribution
    1. Properties of Normal r.v.'s
    2. Linear Combinations of Normal r.v.'s
   E. Related Distributions
    1. Chi-square distribution
    2. t distribution
    3. F distribution 
    4. Relationships among these Distributions
II.  One-Sample and Two-Sample Models
   A. Single-Sample Normal-Data Model
    1. Sample Variance and its Distribution
    2. Role of t distribution
    3. CI for mu and proper interpretation
    4. Hypothesis test (t-test) about mu
      a. Test statistic 
      b. Alternative Hypotheses
      c. Rejection rules and P-value
      d. Proper conclusion for hypothesis test
    5. Connection between CI and Two-Sided Tests
   B. Paired-Samples Normal-Data Model
    1. Finding the differences
    2. Connection to One-Sample Inference
    3. Correct Interpretation of CI for Mean Difference
   C. Two-Independent-Samples Normal-Data Model
    1. Equal-Variance Situation
      a. CI for mu_1 - mu_2
      b. Test statistic and its distribution
      c. Alternative Hypotheses for test
    2. Unequal-Variance Situation
      a. CI for mu_1 - mu_2
      b. Test statistic 
      c. Alternative Hypotheses for test
    3. Correct Interpretation of CI for Difference of Means
   D. Applicability of t-procedures
    1. Requirements for t-procedures
    2. Robustness of t-procedures
    3. Large-sample-size situation
    4. Checking assumptions
    5. Normal Q-Q plots (and other plots)
   E. Nonparametric Alternatives
    1. Sign test
    2. Wilcoxon Signed-Rank Test
    3. Wilcoxon Rank-Sum Test
    4. In what situation should each of these tests be used?
    5. What are the data assumptions for each of these tests?
III. Simple Linear Regression Model
   A. Basics of the SLR Model
    1. Statistical Relationship between Y and X
    2. Role of Response and Predictor in SLR Model
    3. Mathematical Equation and Error Assumptions for SLR Model
    4. Deterministic and Random Component
    5. Mean Response and Variance of Response
    6. Model Using Matrix Notation
   B. Estimation of beta_0 and beta_1
    1. Idea behind Least Squares Method
    2. The "Normal Equations" and the Least Squares Estimators b_0 and b_1
    3. Properties of Least Squares Estimators
   C. Fitted Values and Residuals
   D. Interpreting Predicted Values and Estimated Slope
   E. Estimating the Error
   F. Normal Error Assumption
    1. Why do we need to assume normality?
   G. Inference in SLR Model
    1. Sampling distribution of estimated slope b_1
    2. CI for true slope beta_1
    3. t-test about true slope beta_1
    4. What does this test tell us about the relationship between Y and X?
   H. Inference about the Response Variable
     1. CI for the Mean Response at a particular X-value
     2. PI for the Response of a New Observation at a particular X-value
     3. How are these two intervals different?
     4. Which should be wider?
   I. Analysis of Variance Approach
     1. SSTO, SSR, and SSE
     2. "Partitioning" the Sample Variation in Y
     3. ANOVA table
     4. Reasoning behind the F-test about beta_1
     5. Test statistic & procedure for the F-test for the slope beta_1 
     6. Idea of Reduced and Full Models
   J. Measuring the Linear Relationship between Y and X
     1. Definition of Coefficient of Determination R^2
     2. Properly Interpreting a value of R^2
     3. Definition of Correlation Coefficient r
     4. Properly Interpreting a value of r
IV.   Miscellaneous Regression-Related Topics
   A. Correlation Models
    1. Key Difference between regression model and correlation model
    2. Bivariate normal model
    3. Population correlation coefficient rho
    4. Testing whether rho = 0
    5. Large-sample CI for rho
   B. Cautions about Regression
    1. Predicting Values into the Future
    2. Extrapolation and its Associated Dangers
    3. Does linear association between Y and X imply causation?
    4. Concerns with simultaneous multiple predictions/inferences
    5. Effect of Measurement Error in the X variable(s)
V.  Introduction to Multiple Linear Regression (MLR)
   A. General MLR model with k predictors
    1. Interpretions of regression coefficients in the MLR model
    2. Meaning and Examples of General Linear Model
    3. General Linear Model in Matrix Terms
      a. Y vector
      b. X matrix
      c. beta vector
      d. epsilon vector
    4. Fitting the MLR (estimating the beta's)
      a. vector of estimated coefficients
      b. vector of fitted values
      c. vector of residuals
      d. Interpretations of estimated regression coefficients
   B. Analysis of Variance in MLR
    1. SSTO, SSR, SSE
    2. Degrees of Freedom for each SS
    3. Overall ANOVA F-test
      a. Null and alternative hypotheses
      b. Test statistic value
    4. Coefficient of Multiple Determination R^2
    5. Adjusted R^2
   C. Inference about Individual Regression Coefficients
    1. CI for an individual beta
    2. Test for whether an individual beta = 0
      a. Tests marginal effect of individual predictor
      b. "in the presence of" other predictors in the model
      c. Bonferroni and Holm corrections for multiple tests
   D. CI for the mean response, E(Y_h)
   E. Prediction Interval for 'new' response value, Y_h(new)