STAT 512 -- EXAM 2 REVIEW SHEET 

I.  Sampling Distributions when we have Normal Data
*** NOTE: Topics A, B, C in Section I are kind of necessary background knowledge for Test 2.
*** They will not be asked about directly, but you should know/remember the key results 
*** related to these sampling distributions.
  A.  Sampling Distribution of the Sample Mean Y-bar
     1. Expected Value of Y-bar
     2. Variance of Y-bar
     3. Proof of Normality of Y-bar (when data are normal)
     4. Finding probabilities about Y-bar
  B.  The Chi-square Distribution
     1. Relationship to gamma distribution
     2. Sum of independent chi-square r.v.'s
     3. Other theorem relating two chi-square r.v.'s
  C.  Sampling Distribution of the Sample Variance S^2
     1. When are Y-bar and S^2 independent r.v.'s?
     2. Distribution of (n-1)S^2/(sigma^2) when data are normal
     3. Finding probabilities about S^2
  D.  The t Distribution
     1. Definition of a t r.v. (in terms of normal and chi-square r.v.'s)
     2. "Degrees of freedom' for a t r.v.
     3. Expected value of the reciprocal of a chi-square r.v.
     4. Expected value and variance of a t r.v.
     5. Relationship/comparison of the t and Z (std. normal) distributions 
  E.  Relationship between Y-bar and the t-distribution
     1. Sampling Distribution of the (Y-bar - mu)/[S/sqrt(n)]
     2. Distribution of (n-1)S^2/(sigma^2) when data are normal
     3. Finding probabilities about Y-bar when sigma^2 is unknown
  F.  The F Distribution
     1. Definition of a F r.v. (in terms of two indep. chi-square r.v.'s)
     2. Numerator and Denominator "Degrees of freedom' for an F r.v.
     3. Expected value and variance of an F r.v.
     4. Other properties of F r.v.'s
     5. Finding probabilities about the ratio of two sample variances
II. The Central Limit Theorem
  A.  Precise Formal Statement of the Central Limit Theorem
  B.  Implications of the Central Limit Theorem
     1. Behavior of Y-bar when the sample size is large
     2. Finding probabilities about Y-bar when the sample size is large
     3. Rules of Thumb about how large n should be to apply the CLT
III. The Normal Approximation to the Binomial
  A.  Facts about Binomial Experiments
     1. Definition of p-hat in a binomial experiment
     2. Approximate Distribution of p-hat when n is "large"
     3. Approximate Distribution of # of successes Y when n is large
  B.  Using Normal Probability to approximate a Binomial probability
     1. Correct mean and variance of the normal distribution
     2. Correct use of the Continuity Correction
     3. Rule of thumb for when the Normal approximation works well
  C.  Normal approximation to other Discrete Distributions (e.g., Poisson)
IV.  Estimation
  A.  Basic Ideas
     1. Definition of target parameter
     2. Point estimate and Interval estimate
     3. Estimate vs. Estimator
  B.  Judging Quality of Point Estimators
     1. Performance across repeated samples
     2. Accuracy vs. Precision
  C.  Bias and MSE of a Point Estimator
     1. Bias of an estimator
     2. Unbiased estimator
     3. MSE of an estimator
     4. MSE as a function of bias and variance of the estimator
     5. Comparing candidate estimators based on their MSE
  D.  Common Unbiased Estimators
     1. Y-bar as an estimator of mu
     2. Variance of Y-bar
     3. p-hat as an estimator of p
     4. Variance of p-hat
     5. (Y1-bar - Y2-bar) as an estimator of (mu1 - mu2)
     6. Variance of (Y1-bar - Y2-bar)
     7. (p1-hat - p2-hat) as an estimator of (p1 - p2)
     8. Variance of (p1-hat - p2-hat)
     9. Sample variance S^2 as an estimator of sigma^2
  E.  Estimation Error Bounds
     1. Definition of error in estimation
     2. Empirical Rule
     3. Using empirical rule to get 95% bound on estimation error
     4. Chebyshev's rule for bound on estimation error of non-normal estimators
V.  Confidence Intervals
  A.  Properties of Interval Estimators
     1. Confidence Coefficient
     2. Two Sided Intervals, Lower Confidence Bounds, Upper Confidence Bounds
  B.  Pivotal Method
     1. Definition of Pivotal Quantity
     2. Using Distribution of Pivotal Quantity to set up a Probability Statement
     3. Solving Inequality for Parameter to get CI formula
  C.  Large-Sample Confidence Intervals
     1. General Formula for CIs when unbiased estimator has approximately normal sampling distn
     2. Meaning of confidence level
     3. Examples of Large-sample CIs
       a. CI for mu
       b. CI for p
       c. CI for mu1 - mu2
       d. CI for p1 - p2
     4. Interpreting CIs in the context of the variable(s) in the problem
  D.  Sample Size Determination Formulas