Midterm III Review Sheet for Math 156.6, Fall 2008

Words/concepts you should know (i.e.,their definitions, examples, their relationships,...)

1. The Central Limit Theorem (CLT)
   1. Its statement:
   2. Using it to calculate (approximate) probabilities for sample means when something is known about the population mean and standard deviation (even if the population distribution is not particularly Normal).
2. Investigating a population mean (μ)
   1. Confidence intervals for means when σ is known:
      1. the formula
      2. the meaning of the confidence level (IMPORTANT)
      3. the meaning of "margin of error" (MofE)
      4. how to choose a sample size to get a particular MofE at a particular confidence level
   2. Hypothesis testing about means when σ is known:
      1. what are the "null hypothesis" (H₀) and the "alternate hypothesis" (H_a)
      2. the three standard forms for H_a — "one-tailed" v.s. "two-tailed" tests
      3. what are the "test statistic" (the "z-statistic") and its formula
      4. what is the "p-value" for the three different forms of H_a and what does it mean (IMPORTANT)
      5. using the p-value, when do we "reject the null hypothesis" or "fail to reject the null hypothesis", at a given "significance level"
   3. When σ is unknown:
      1. the "standard error" of a statistic (particularly, the statistic x)
      2. confidence intervals:
         1. the formula
         2. the "degrees of freedom" df
      3. hypothesis testing:
         1. the "t-statistic" and its formula
         2. the p-value for each different kind of H_a
      4. conditions for use of the T-distribution
         1. SRS
         2. symmetric
         3. no outliers
         4. single peaked
         5. for which sizes of n are which conditions most important
         6. how to look for these conditions in your sample data

Things to know how to do:

1. Most importantly: the homework problems which were assigned, and similar problems from the same chapters, which may not have been assigned. I.e., make sure you are comfortable with the assigned problems, then look through those sections and try the unassigned next problem after, or the problem right before, some of the assigned ones.
2. Compute a ..
   1. probability for a sample mean in some range when the population mean and standard deviation (and the sample size) are known
   2. confidence interval
   3. margin of error
   4. sample size in order to get a particular MofE
   5. p-value for all tests in all situations
   6. t-statistic (so, also be able to use tcdf in your calculator)
   7. z-statistic
   8. CI or HT for a matched-pairs scenario
3. Discuss (succinctly, accurately and completely)...
   1. the meaning of the confidence level of a confidence interval
   2. the meaning of the p-value of a hypothesis test
   3. the conditions under which it makes sense to use
      1. a confidence interval for a mean with known σ
      2. a hypothesis test for a mean with known σ
      3. a confidence interval for a mean with unknown σ (a T-test)
      4. a hypothesis test for a mean with unknown σ (a T-test)
4. Set up a good solution of a test of significance problem, following the standard steps:
   1. Define the RV and the μ which are are the subject of the test. Then define the null and alternate hypotheses to which you will apply the test.
   2. Compute the statistic (a z-statistic for the Z-test, i.e., a test with known population standard deviation σ; a t-statistic for the T-test, i.e., a test with unknown population standard deviation σ).
   3. Compute the p-value.
   4. Draw your conclusion -- i.e., either "we reject the null hypothesis" or not ... and then restate this conclusion in the terms in which the problem was originally posed.