Multiple choice questions;

Concept 1. Which of the following represents a valid probability table? (i) outcomes 1 2 3 4 5 probability 1/5 1/5 1/5 1/5 1/5 (ii) outcomes 1 2 3 5 probability 1/2 1/5 1/10 1/10 1/10 Circle the best choice: A. (i) B. (ii) C. (i) and (ii) D. Not enough information Concept 2. True or false: Setting the prior probability of a hypothesis to 0 means that no amount of data will make the posterior probability of that hypothesis the maximum over all hypotheses. Circle one: True False Concept 3. True or false: It is okay to have a prior that depends on more than one unknown parameter. Circle one: True False Concept 4. Data is drawn from a normal distribution with unknown mean p. We make the following hypotheses: Ho: =1 and HA: > 1. For (i)-(ii) circle the correct answers: (i) Is Ho a simple or composite hypothesis? Simple Composite (ii) Is HA a simple or composite hypothesis? Simple Composite (iii) Is HA a one or two-sided? One-sided Two-sided Concept 5. If the original data has n points then a bootstrap sample should have A. Fewer points than the original because there is less information in the sample than in the underlying distribution. B. The same number of points as the original because we want the bootstrap statistic to mimic the statistic on the original data. C. Many more points than the original because we have the computing power to handle a lot of data. Circle the best answer: A B C. Concept 6. In 3 tosses of a coin which of following equals the event "exactly two heads"? A = {THH, HTH, HHT, HHH} B = {THH, HTH, HHT} C = {HTH, THH} Circle the best answer: A B C B and CConcept 7. These questions all refer to the following figure. For each one circle the best answer. (i) The probability a represents A. P(A,) B. P(A,[B,) C. P(B,JA,) D. P(Ci|B, n A, ). (ii) The probability y represents A. P(B,) B. P(A,|B,) C. P(B,|4,) D. P(Ci|B, n A,). (iii) The probability = represents A. P(C,) B. P(BIC,) C. P(CB2) D. P(Ci|B, n A,). (iv) The circled node represents the event A. C, B. B, n C, C. An B, n C, D. GB, n A. Concept 8. The graphs below give the pmf for 3 random variables. (A) (B) (C) Circle the answer that orders the graphs from smallest to biggest standard deviation. ABC ACB BAC BCA CAB CBA Concept 9. Suppose you have $100 and you need $1000 by tomorrow morning. Your only way to get the money you need is to gamble. If you bet Sk, you either win $k with probability p or lose Sk with probability 1 - p. Here are two strategies: Maximal strategy: Bet as much as you can, up to what you need, each time. Minimal strategy: Make a small bet, say $10, each time. Suppose p = 0.8. Circle the better strategy: Maximal 2. Minimal Concept 10. Consider the following joint pdf's for the random variables X and Y. Circle the ones where X and Y are independent and cross out the other ones. A. f(x,y) = Arly' B. f(x,y) = 1(ay + zy'). C. f(x, y) = 6e-3x-24Suppose A ~ Bernoulli(0) where o is unknown. Which of the following is the correct statement? A. The random variable is discrete, the space of hypotheses is discrete. B. The random variable is discrete, the space of hypotheses is continuous. C. The random variable is continuous, the space of hypotheses is discrete. D. The random variable is continuous, the space of hypotheses is continuous. Circle the letter of the correct statement: A B C D Concept 12. Let o be the probability of heads for a bent coin. Suppose your prior f() is Beta(6, 8). Also suppose you flip the coin 7 times, getting 2 heads and 5 tails. What is the posterior pdf f(Or)? Circle the best answer. A. Beta(2,5) B. Beta(3,6) C. Beta(6,8) D. Beta(8,13) E. Not enough information to say Concept 13. Suppose the prior has been set. Let z, and r2 be two sets of data. Circle true or false for each of the following statements. A. If r, and ry have the same likelihood function then they result in the same posterior. True False B. If r, and ra result in the same posterior then they have the same likelihood function. True False C. If z, and x2 have proportional likelihood functions then they result in the same True False posterior. Concept 14. Each day Jane arrives X hours late to class, with X ~ uniform(0, 0). Jon models his initial belief about 6 by a prior pdf f(). After Jane arrives a hours late to the next class, Jon computes the likelihood function f(r|0) and the posterior pdf f(ex). Circle the probability computations a frequentist would consider valid. Cross out the others. A. prior B. posterior C. likelihood Concept 15. Suppose we run a two-sample t-test for equal means with significance level a = 0.05. If the data implies we should reject the null hypothesis, then the odds that the two samples come from distributions with the same mean are (circle the best answer) A. 19/1 B. 1/19 C. 20/1 D. 1/20 E. unknown Concept 16. Consider the following statements about a 95% confidence interval for a parameter e. A. P(60 is in the CI | 0 = 60) 2 0.95 B. P(60 is in the CI ) 2 0.95 C. An experiment produces the CI [-1, 1.5): P(0 is in [-1, 1.5] | 0 = 0) 2 0.95 Circle the letter of each correct statement and cross out the others: B C