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Sampling 101
Estimating with Confidence
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I regress...
Bayes and Nonparametrics
100
The three quartiles, mean, and standard deviation of the data set 23, 17, 30, 19, 26.
What are 18, 26, 28, 23, and 5.24. Follow-up: How could you display this data graphically?
100
This is the method of moments estimator for p, when p is the unknown probability of a coin landing hands, and you observe n coin flips.
What is Xbar, the proportion of heads observed in the n flips (see the Bernoulli question from Quiz 2).
100
A 95% confidence interval for the mean of a population is found to be (43.5, 47). This is the p-value of a two-tailed significance test from this sample testing whether the mean is equal to 47.
What is 0.05? (Recall that the p-value is the smallest significance level leading to a rejection of H0, and that for a two-tailed test we reject H0 at level alpha if and only if the (1-alpha) confidence interval does not contain the value in the null hypothesis.) Followup: Prove this.
100
I can find estimators for any parameters in a regression equation because the least squares regression equation is defined as this.
What is the equation which minimizes the sum of squared errors? (Be sure you can derive all the formulas [see for example Quiz 3] in addition to being able to use them)
100
This is the difference between the classical (frequentist) approach to statistical inference and the Bayesian approach.
What is the classical approach assumes no prior knowledge about the unknown parameter and views the sampling procedure as the only source of randomness, whereas the Bayesian approach allows a prior distribution to be placed on the unknown parameter, and inferences can be made based on both that distribution and the sample data?
200
The sum of the square of n-1 standard normal random variables has this distribution, as does (n-1)S^2/sig^2 when S^2 is a sample variance from n-1 observations of a population with variance sig^2.
What is chi-square with n-1 degrees of freedom? (See Theorems 4.2.7 and 4.2.8 in the text, and also our class notes.)
200
Its abbreviation is MLE, and this is the intuition behind the MLE as a good estimate for an unknown parameter.
What is maximum likelihood estimator? We compute how likely it is to observe the sample that we have observed, and choose the parameter that maximizes this liklihood.
200
This is the Neyman-Pearson Lemma.
What is a statement which says that the most powerful test has a rejection region derived from the ratio of the liklihood functions evaluated at the null and alternative parameters. (See Theorem 7.2.1)
200
A residual plot is used for this.
What is to assess how well the least squares regression equation fits the data? The plot of the residuals should not have any trends, the residuals should be reasonably small and appear randomly distributed.
200
This is the definition and formula for the posterior distribution of an unknown parameter theta.
What is the distribution of theta, conditioned on the sample data, pi(theta | x) = f(x|theta)pi(theta)/g(x)?
300
The ratio of two sample variances has this distribution.
What is the F-distribution?
300
What happens to the margin of error in a confidence interval as the sample size n goes to infinity?
What is it goes to zero?[Can you prove this for a normally distributed population? Hint: What happens to the standard error?]
300
Suppose two independent samples are taken from normal populations of sizes 25 and 16. This is the test you would use to test whether the two population variances are equal.
What is the F-test? (See Section 7.5 to review this test)
300
When the random error in a least squares regression model is assumed to be normally distributed, the distribution of the estimators to the slope m and intercept b is this.
What are normally distributed with (for m:) mean m and variance sigma^2/Sxx, and (for b:) mean b and variance (1/n + xbar^2/Sxx)sigma^2? (See Theorem 8.2.1)
300
These are the Bayes point estimates under quadratic and absolute loss.
For quadratic, what is the estimator which minimizes the square of the error (which happens to also be the expected value of the posterior distribution), and the estimator which minimizes the absolute value of the error? See Section 11.2.1. Follow up: Prove that the estimator which minimizes the quadratic loss is the usual Baye's point estimator (the expected value of the posterior)
400
We can approximate a binomial random variable by a normal random variable because of *this theorem* and using *this* type of correction.
What is the Central Limit Theorem and the continuity correction?
400
This is the definition of a pivotal quantity for an unknown parameter t, and this is the pivotal quantity for mean mu of a normal distribution.
What is a function of t whose distribution does not depend on t? For mu of a normal distribution X, a pivotal quantity is the standardized random variable (X - mu)/(sigma/sqrt(n)). See Section 6.1 (and class notes) for more examples of pivotal quantities.
400
This test is used when testing for independence between two variables or in testing whether the probabilities of a multinomial distribution are specified values.
What is the Pearson Chi-square test?
400
When performing hypothesis tests about the estimators to slope an intercept parameters m and b for a data set, the variance sigma^2 of the random error can be approximated by this value.
What is the mean square error, MSE = SSE/(n-2)? (See Section 8.2.5)
400
If the population median is M, and N is the number out of a sample of observations which lie below M, this is the distribution of N. In contrast, if the median of two samples is m, and N' is the number of observations out of the sample from population 1 that lie below m, this is the distribution of N'.
What is binomial (for N) and hypergeometric (for N')?
500
If X1, X2, ..., Xn are iid observations of a continuous random variable and we sort them from lowest to largest to obtain Y1, Y2, ..., Yn, the random variables Yi are called this.
What is the order statistics of the random sample X1, ..., Xn? (See Section 4.3 and review how to find the distributions of these random varianbles)
500
This is the correct interpretation of a 95% confidence interval of (4.5, 6.7) for the mean mu of a distribution.
What is: We are 95% confident that the true mean mu lies between 4.5 and 6.7. This means that if we repeatedly took samples and created such intervals that 95% of the time our interval created from the sample will contain the true mean mu.
500
Suppose we have a random sample of size 25 from a normal population with an unknown mean μ and known standard deviation. We wish to test the hypothesis H0 : μ = 10 vs. Ha : μ > 10. This is how you would find the power of this test against the alternative that μ = 11.
What is: find the rejection region of the test, find the probability of making a type II error (ie. failing to reject when we should have) if indeed μ = 11, 1 minus this value is the power.
500
This transform is used when testing the correlation between two variables, before doing regression analysis.
What is the Fisher-z transform, 0.5log((1+r)/(1-r))? (See Section 8.5)
500
The difference between the Signed test and Wilcoxon signed rank test.
What is: the signed test counts the number of observations below the proposed median and rejects that median if this number is too large or small, whereas the Wilcoxon test also takes the magnitudes of the observations into account, ranking their values and summing the ranks of those observations which are below the proposed median. If this number is too small or too large, we reject the null hypothesis (for two-sided tests).