STAT 515 -- Review Sheet, Final Exam 

I. Hypothesis Tests
 A.  Purpose of Hypothesis Testing
   1. Null and alternative hypotheses
     a.  Which one is "assumed" to be true before the test?
     b.  Which one are we "trying to prove" (seeking evidence for)?
   2. Type I error
   3. Significance level (alpha) for a hypothesis test
   4. How do we decide whether to reject H_0 (whether result is "significant")?
 B.  Hypothesis Test About a Proportion
   1. Test Statistic z
     a. What distribution does z have if H_0 is true?
   2. Determining rejection region (for each type of alternative)
   3. Stating conclusions in terms of the variables in the problem 
 C.  P-values
   1. Precise definition of P-value
   2. What does p-value tell us about the evidence against H_0?

II. Inference in Two-Sample Situations
 A. Paired Samples
   1. What kinds of studies yield paired data? (Examples)
   2. Key characteristic of paired samples
   3. Performing Inference about the Mean of the Differences
   4. t-test about mu_d
   5. CI about mu_d
   6. How does the paired-data problem reduce to a one-sample problem?
 B.  Independent Sample Problems
   1.  Key difference between Independent Samples and Paired Samples
   2.  Comparing Two Population Means
     a. Assumptions
     b. Equal-variance case and unequal-variance case
   3.  CI for mu_1 - mu_2
   4.  t-test of H_0: mu_1 = mu_2
     a. test statistic t
     b. Determining correct degrees of freedom
     c. Finding rejection region and making correct conclusion
   5.  Comparing Two Proportions
     a.  Sampling distribution of p_1-hat - p_2-hat
     b.  Assumptions for the approximate normality to be valid
   6.  Hypothesis test of p_1 = p_2
     a. test statistic z
     b. pooled sample proportion p-hat
     c. Finding rejection region and making correct conclusion

III.  Analysis of Variance (ANOVA) with a Completely Randomized Design
  A. Basic Terms 
   1.  Designed Experiment vs. Observational Experiment
   2.  Response Variable
   3.  Factors (Quantitative and Qualitative)
   4.  Levels of a Factor
   5.  Treatments
   6.  Experimental Units
  B.  Completely Randomized Design
   1. Hypothesis Test for whether all treatment means are equal
   2. Comparing variance WITHIN groups to variance BETWEEN groups
   3. SST, SSE, MST, MSE (What do these quantities measure?)
   4. The ANOVA F-statistic:  F = MST/MSE
     a. What is its distibution if H_0 is true?
     b. How do we use it to test whether all treatment means are equal?
   5. Summarizing the data with an ANOVA table
   6. Assumptions of ANOVA F-test
   7. Rejection region and proper conclusions for ANOVA F-test

IV.  Regression Analysis
  A. Probabilistic vs. Deterministic Relationship
  B. (Straight-Line) Simple Linear Regression Model
    1. Response Variable Y, Predictor Variable X
    2. Y-intercept beta_0, Slope beta_1, random error epsilon
  C. Fitting the Model with least squares
    1. Determining whether a straight-line model is appropriate
    2. Scatterplot
    3. Least squares philosophy (minimizing SSE)
    4. The least-squares estimates for beta_0 and beta_1
    5. Interpreting estimated slope and estimated Y-intercept
    6. Using least-squares line to predict Y values for a given X 
    7. Extrapolation
  D. Model Assumptions
    1. mean of random error component = 0
    2. variance of random error component constant for all values of X
    3. Probability distn. of random error component is normal
    4. Values of random error component for any two Y-values are independent
  E. Estimating the error variance sigma^2
    1. s^2 = MSE
    2. s = sqrt(MSE) (estimate of sigma)
    3. Interpretation of s
  F.  Testing the Usefulness of the Model
    1. How does testing whether the slope is 0 test the usefulness of the model?
    2. Test statistic, rejection region for test of H_0: beta_1 = 0
    3. Confidence Interval for the true slope beta_1
  G. Correlation and Coefficient of Determination
   1. Linear Correlation
     a. What it measures and how it is interpreted
     b. Type of variables for which correlation can be measured
     c. Linear association vs. Curved association
   2. r-squared and its interpretation
  H. Estimation and Prediction with the Regression Model
   1. CI for the mean of Y at a particular X value
   2. Prediction Interval for a new Y at a particular X value
   3. Which of these intervals is wider?

V. Inference for Categorical Data
 A.  Analysis of One-Way Tables
   1. Multinomial Data
     a.  How is this an extension of binomial data?
     b.  Characteristics of a multinomial experiment.
   2. Arranging Cell Counts into a One-Way Table
   3. Hypothesis Test about Multinomial Probabilities
     a. Null and alternative hypotheses
     b. How to find Test Statistic
     c. How do we decide whether to reject H_0?
     d. Finding critical value from chi-square table (Table VII) (d.f.?)
     e. Rule of Thumb for checking large-sample assumption
     f. Correct conclusion
 B.  Analysis of Two-Way Tables
   1. Arranging cell counts in Two-Way (Contingency) Table
   2. Notation:  Observed Cell Counts, Row Totals, Column Totals
   3. Notation:  Cell probabilities, marginal probabilities (row & column)
   4. Finding expected cell counts under hypothesis of independence
   5. Chi-square test for independence
     a. Null and alternative hypotheses
     b. How to find Test Statistic
     c. How do we decide whether to reject H_0?
     d. Finding critical value from chi-square table (Table VII) (d.f.?)
     e. Rule of Thumb for checking large-sample assumption   
     f. Correct conclusion