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 Orange – Nonparametric Procedures and

Categorical Data Analysis

BLUE 1

Suppose you have an estimator theta-hat, and you want to know its bias. How is bias computed?

BLUE 2

How is MSE of an estimator computed?

BLUE 3

What is a common unbiased point estimator for a population mean and what is its standard error?

BLUE 4

What is a common unbiased point estimate of a difference in two population proportions, and what is its standard error?

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

What are the 2 properties of pivot quantities and what are pivots used for?

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

How is a t distribution formed?

BLUE 9

How is an F distribution formed?

BLUE 10

How do you form a small-sample confidence interval for a population mean?

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

What is the definition of consistency for an estimator?

Bonus: What concept of convergence is this equivalent to?

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

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

What is the result you can use to show sufficiency without resorting to computing conditional pdfs?

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

Describe how the method of moments works.

PINK 8

Describe how the method of ML estimation works.

PINK 9

A main property of MLEs is that they are _____________, which means that ….

PINK 10

If an estimator is NOT admissible (i.e. inadmissible), what does that mean?

Give an example of an inadmissible estimator.

YELLOW 1

What is the difference between simple and composite hypotheses?

YELLOW 2

Describe the relationships between the two types of error in a hypothesis test, as well as their connection to power.

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

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

How are hypothesis tests and confidence intervals related?

YELLOW 6

What is the difference between the pooled and unpooled t-tests for 2 independent samples when considering tests for means?

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

What does the Neyman-Pearson Lemma say?

(Get the gist of it, what does it let you find, and how?)

YELLOW 9

How do you determine if a most powerful test is UMP?

YELLOW 10

How do you construct a likelihood ratio test?

What is the asymptotic distribution related to LRTs?

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

What is the difference between a proper and improper prior?

What is the difference between an informative and uninformative prior?

BROWN 3

How do you find the posterior density of theta?

BROWN 4

What are conjugate priors?

Give an example of a conjugate prior.

BROWN 5

How would you find the Bayes estimate of: theta theta(1-theta)

if you had the posterior density of theta?

BROWN 6

A Bayes estimator is ALWAYS a function of a _______________ statistic because of the _______________ ________________.

BROWN 7

How is a Bayesian credible interval different from a Frequentist confidence interval?

BROWN 8

Is it possible for Bayesian and Frequentists intervals to agree? If yes, how might this happen?

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

What are some of the issues related to working with Bayes’ factors?

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

When first developing regression models, this is the only constraint on the error terms.

GREEN 3

If your regression model was:

Then how many parameters do you need to estimate?

3322110)( xxxYE

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

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

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

Why do we end up using a t distribution for inference about slope parameters in regression instead of a normal distribution?

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

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

What is correlation and how do we test about it?

ORANGE 1

Describe the two-sample shift model.

ORANGE 2

Describe how the sign test works.

ORANGE 3

Describe how the signed rank test works.

ORANGE 4

Describe how the Wilcoxon Rank Sum/Mann-Whitney U test works.

ORANGE 5

How does a Kolmogorov-Smirnov one-sample test work? Is the null hypothesis in the procedure simple or composite?

ORANGE 6

How does the 2-sample Kolmogorov-Smirnov test work?

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

How is a chi-square goodness of fit test performed? When should you perform one?

ORANGE 9

How (and when) does a chi-square test of independence work?

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.

Takehome Final Exam is due Friday, May 13th at 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!

REMINDER:

Math dept. end of semester picnic is Saturday from 12-2 at the Alumni House

THANKS FOR A GREAT SEMESTER!