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Math Stat Trivial Pursuit (Sort of) For Review (math 30). Colors and Categories. Blue – Basics of Estimation Pink – Properties of Estimators and Methods for Estimation Yellow – Hypothesis Testing Brown – Bayesian Methods Green – Regression

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colors and categories
Colors and Categories
  • Blue – Basics of Estimation
  • Pink – Properties of Estimators and Methods for Estimation
  • Yellow – Hypothesis Testing
  • Brown – Bayesian Methods
  • Green – Regression
  • Orange – Nonparametric Procedures and Categorical Data Analysis
blue 1
Blue 1
  • Suppose you have an estimator theta-hat, and you want to know its bias. How is bias computed?
blue 2
Blue 2
  • How is MSE of an estimator computed?
blue 3
Blue 3
  • What is a common unbiased point estimator for a population mean and what is its standard error?
blue 4
Blue 4
  • What is a common unbiased point estimate of a difference in two population proportions, and what is its standard error?
blue 5
Blue 5
  • A very important result related to samples from a normal distribution is that:
    • The sample mean is ____________ distributed.
    • The sample variance, appropriately scaled, is ____________ distributed.
    • The sample mean and sample variance are ____________________.
  • (Fill-in all three blanks for credit).
blue 6
Blue 6
  • What are the 2 properties of pivot quantities and what are pivots used for?
blue 7
Blue 7
  • How would you use the asymptotic normal distribution of many unbiased point estimators to create a confidence interval for their respective parameters?
  • (You can just give the formula).
  • Hint: Think of a specific case and generalize.
blue 8
Blue 8
  • How is a t distribution formed?
blue 9
Blue 9
  • How is an F distribution formed?
blue 10
Blue 10
  • How do you form a small-sample confidence interval for a population mean?
pink 1
Pink 1
  • If relative efficiency is computed between two estimators, it means that both estimators were _______________, and if the numerical value of the relative efficiency is 2, then it means that the _____________ (first or second) estimator is better.
pink 2
Pink 2
  • What is the definition of consistency for an estimator?
  • Bonus: What concept of convergence is this equivalent to?
pink 3
Pink 3
  • For an unbiased estimator, what is the “fast” way of showing consistency?
  • Bonus: Do you remember what convergence result this was derived from?
pink 4
Pink 4
  • If you have a RS of n observations from a distribution with unknown parameter theta, and T is sufficient for theta, what does that mean?
pink 5
Pink 5
  • What is the result you can use to show sufficiency without resorting to computing conditional pdfs?
pink 6
Pink 6
  • What does the Rao-Blackwell Theorem say?
  • Bonus: What’s the fast way of finding the quantity RB refers to in the end?
pink 7
Pink 7
  • Describe how the method of moments works.
pink 8
Pink 8
  • Describe how the method of ML estimation works.
pink 9
Pink 9
  • A main property of MLEs is that they are _____________, which means that ….
pink 10
Pink 10
  • If an estimator is NOT admissible (i.e. inadmissible), what does that mean?
  • Give an example of an inadmissible estimator.
yellow 1
Yellow 1
  • What is the difference between simple and composite hypotheses?
yellow 2
Yellow 2
  • Describe the relationships between the two types of error in a hypothesis test, as well as their connection to power.
yellow 3
Yellow 3
  • If you have a test statistic, you can use either a rejection region approach or a p-value approach to determine if the null hypothesis should be rejected. What is the difference in the 2 approaches? (Describe).
yellow 4
Yellow 4
  • For the common large sample asymptotically normal z-tests, what is the rejection region for a 2-tailed test?
  • Bonus: If the significance level is .05 for this test, what is the range of test statistics where you would NOT reject the null hypothesis (numerical values).
yellow 5
Yellow 5
  • How are hypothesis tests and confidence intervals related?
yellow 6
Yellow 6
  • What is the difference between the pooled and unpooled t-tests for 2 independent samples when considering tests for means?
yellow 7
Yellow 7
  • In order to determine which 2-sample t-test for small sample sizes is appropriate, you might have to run a test to check for equality of _______________, and in order to control your overall significance level, you might have to use a ____________ _____________.
yellow 8
Yellow 8
  • What does the Neyman-Pearson Lemma say?
  • (Get the gist of it, what does it let you find, and how?)
yellow 9
Yellow 9
  • How do you determine if a most powerful test is UMP?
yellow 10
Yellow 10
  • How do you construct a likelihood ratio test?
  • What is the asymptotic distribution related to LRTs?
brown 1
Brown 1
  • What is the major difference between Frequentist and Bayesian approaches to statistics in terms of how the parameter theta is treated?
brown 2
Brown 2
  • What is the difference between a proper and improper prior?
  • What is the difference between an informative and uninformative prior?
brown 3
Brown 3
  • How do you find the posterior density of theta?
brown 4
Brown 4
  • What are conjugate priors?
  • Give an example of a conjugate prior.
brown 5
Brown 5
  • How would you find the Bayes estimate of:
    • theta
    • theta(1-theta)

if you had the posterior density of theta?

brown 6
Brown 6
  • A Bayes estimator is ALWAYS a function of a _______________ statistic because of the _______________ ________________.
brown 7
Brown 7
  • How is a Bayesian credible interval different from a Frequentist confidence interval?
brown 8
Brown 8
  • Is it possible for Bayesian and Frequentists intervals to agree? If yes, how might this happen?
brown 9
Brown 9
  • Bayesian hypothesis testing is performed using ______ ________, which are Bayesian analogues of ________ test procedures, and which can allow you to find evidence in favor of your ___________ hypothesis.
brown 10
Brown 10
  • What are some of the issues related to working with Bayes’ factors?
green 1
Green 1
  • Relationships between two variables, X and Y can be deterministic or ________________. Regression is used when the relationship is _______________. This means that ….
green 2
Green 2
  • When first developing regression models, this is the only constraint on the error terms.
green 3
Green 3
  • If your regression model was:
  • Then how many parameters do you need to estimate?
green 4
Green 4
  • In least squares solutions for regression, what quantity is minimized to find the solution?
  • (You can just give the simple LR quantity).
green 5
Green 5
  • The least squares estimates are all ____________, and their variances are functions of _____________, which in turn can be estimated by _______, which is equal to (1/(n-2))SSE.
green 6
Green 6
  • What is the full set of conditions on the error terms in order to get normal sampling distributions for the parameter estimates if sigma is known?
green 7
Green 7
  • Why do we end up using a t distribution for inference about slope parameters in regression instead of a normal distribution?
green 8
Green 8
  • What is the main difference between a confidence interval for a mean response and a prediction interval for an individual response in regression?
green 9
Green 9
  • How are CIs for mean responses and prediction intervals for individual responses affected as the chosen x moves further from the mean of the x’s?
green 10
Green 10
  • What is correlation and how do we test about it?
orange 1
Orange 1
  • Describe the two-sample shift model.
orange 2
Orange 2
  • Describe how the sign test works.
orange 3
Orange 3
  • Describe how the signed rank test works.
orange 4
Orange 4
  • Describe how the Wilcoxon Rank Sum/Mann-Whitney U test works.
orange 5
Orange 5
  • How does a Kolmogorov-Smirnov one-sample test work? Is the null hypothesis in the procedure simple or composite?
orange 6
Orange 6
  • How does the 2-sample Kolmogorov-Smirnov test work?
orange 7
Orange 7
  • When performing categorical data analysis, the main distribution you need to understand for the theoretical setup of problems is the ______________ distribution, but the test statistics turn out to have a different distribution, which is the ________________ distribution.
orange 8
Orange 8
  • How is a chi-square goodness of fit test performed? When should you perform one?
orange 9
Orange 9
  • How (and when) does a chi-square test of independence work?
orange 10
Orange 10
  • For 2x2 tables, inference is also possible with:
    • _________ exact test for small sample sizes
    • _________ ratios, which relies on an asymptotic ______ distribution for it’s natural log.
reminder
Reminder:
  • Takehome Final Exam is due Friday, May 13that 5 p.m. SHARP.
  • Office Hours (see front cover of exam):
    • Monday 9-12 during my other course’s exam
    • Tuesday 10-12
    • Wednesday 1-3
    • Thursday 1-3
    • Any other time by appt. – just send me an email!
thanks for a great semester
Thanks for a Great Semester!
  • Math dept. end of semester picnic is Saturday from 12-2 at the Alumni House