Quiz 1. Name: Instructions: Closed book, notes, and no electronic devices.

Transcription

1 Quiz 1. Name: Instructions: Closed book, notes, and no electronic devices. 1.(10) What is usually true about a parameter of a model? A. It is a known number B. It is determined by the data C. It is an unknown number D. It is determined by the DATA 2. (10) You flip a coin 10 times, and get 6 heads in those ten flips. You flip it ten more times, and get 5 heads. You repeat this process 1000 times. How many out of the 1000 sets of ten flips will have exactly five heads? A. 50 B. 250 C. 500 D. 750 E (20) Solve the equation R 1 = (Y 1 Y 0)/Y 0 for Y 1. Show your work. 4.(10) What was assumed about the stock returns in the simulation study of Chapter 1? A. 50% are greater than 0. B. They come from a normal distribution C. They are a function of trading volume D. Arbitrage profits exist 5.(10) Pick the best model: A. purely deterministic B. purely probabilistic C. normal distribution D. a model with both deterministic and probabilistic components 6.(10) Why are rules of thumb ugly? A. because they are difficult to implement B. because they are no fun C. because they are not logical facts D. because they are bad advice 7.(10) In the weight example p(80 kg) = Then A. 1.7% of people weight between 79.5 and 80.5 kg B. 1.7% of people weigh kg C. The area under the curve at y=80 kg is D. Weights are normally distributed

2 Quiz 2: Name: Closed books, notes, and no electronic devices. 1. (20) Suppose f (x 0) = 0. Then A. f(x 0) is the minimum value of the function B. f(x 0) is the maximum value of the function C. f(x 0) is either a local minimum value or a local maximum value of the function D. x 0 is either a local minimum value or a local maximum value of the function 4 2. (40) Suppose f ( x) dx 5. Draw a graph that illustrates this fact. Don t calculate anything; just draw the graph (20) Suppose 1000 U* values are produced by the uniform distribution between 0 and 1 (the U(0,1) distribution in the notation of the book). About how many of these 1000 values will be between 0.60 and 0.75? A. 15 B. 150 C. 600 D. 750

3 Quiz 3: Name: Closed books, notes, and no electronic devices. 1.(40) When is the distribution p(y) a "good model"? (One sentence only). 2.(20) Give an example of a specific, "named" distribution p(y). (One or two words only. No math.) 3.(20) The is an estimate of p(y). A. probability distribution function B. cumulative distribution function C. quantile function D. histogram

4 Quiz 4: Name: Closed books, notes, and no electronic devices. Here is the normal q-q plot of the call center data from the book: The data points do not lie on a straight line. As described in the book, how can you tell if the differences between the points and the line is explainable by chance alone? (1-3 sentences maximum).

5 Quiz 5: Name: Closed books, notes, and no electronic devices. Let Y = a person s expense on housing, and let X = that person s income. Define p(y x) in one or two sentences.

6 Quiz 6. Name: Instructions: Closed book, notes, and no electronic devices. Suppose the conditional distributions of Dayta pronunciation, given Gender, are as follows: y p(y Male) Day ta 0.8 Daa ta 0.2 y p(y Female) Day ta 0.6 Daa ta 0.4 Suppose the marginal distribution of Gender is as follows: x p(x) Male 0.3 Female 0.7 Fill in the values of the joint distribution table. Day ta Daa ta Male Female

7 Quiz 7. Name: Closed book, notes, and no electronic devices. From the book, the dashed curve is p(y X=Stealer) and the solid curve is p(y X=Non-Stealer). Here y is an employee fitness score, based on a questionnaire. Suppose the prior distribution of stealing behavior is as follows: x p(x) Stealer.5 Non-Stealer.5 Using the graph, estimate p(x Y=55), the posterior distribution of stealing behavior, given y = 55.

8 Quiz 8. Name: Closed book, notes, and no electronic devices. The following data comprise a population : 1, 3, 1, 4, 10. Give the population distribution p(y).

9 Quiz 9. Name: Closed book, notes, and no electronic devices. The most common assumption in statistics is that the data Y1, Y2,, Yn are produced as an independent and identically distributed (iid) sequence. 1. (40) Briefly give a real example of data Y1, Y2,, Yn that are not independent. 2. (40) Briefly give a real example of data Y1, Y2,, Yn that are not identically distributed.

10 Quiz 10. Name: Closed book, notes, and no electronic devices. Here is a distribution that can produce iid data Y 1, Y 2,, Y n. y p(y) What does the Law of Large Numbers tell you in this case?

11 Quiz 11. Name: Closed book, notes, and no electronic devices. Here is a distribution that produces Y. y p(y) -1 1/4 0 1/2 1 1/4 1.0 Find E(Y 2 ).

12 Quiz 12. Name: Closed book, notes, and no electronic devices. 1. Suppose f(y) = y 1/2, for y>0. Use calculus to identify whether the function is concave, convex, or whether you just can t tell. 2. The mean of the stock returns was.0002 and the standard deviation was The two standard deviation range is thus (.0112), or the range from.0222 to What does Chebychev s theorem tell you about the stock returns and this range of values?

13 Quiz 13. Name: Closed book, notes, and no electronic devices. Let X = husband s income and Y = wife s income (in thousands of dollars). Suppose: Var(X) = 100 (Hence the standard deviation of X is 10) Var(Y) = 64 (Hence the standard deviation of X is 8) Covariance(X,Y) = 40 (Hence the correlation between X and Y is 40/(10*8) = 0.50). Let T = X+Y be the household combined income. Find the variance of T.

14 Quiz 14. Name: Closed book, notes, and no electronic devices. Suppose Y1 and Y2 are produced by a Bernoulli distribution with parameter. You know from the homework that E(Y1) = and that E(Y2) =. Show, step by step, that 2Y1 Y2 is an unbiased estimator of. Give the reason for each step.

15 Quiz 15. Name: Closed book, notes and no electronic devices. When is the median more efficient than the mean?

16 Quiz 16. Name: Closed book, notes and no electronic devices. A bent coin is tossed three times, giving heads, tails, heads. The parameter is = Pr(heads). Give the parameter space.

17 Quiz 17. Name: Closed book, notes and no electronic devices. What assumption is needed about the likelihood function in order to use the Wald standard error?

18 Quiz 18. Name: Closed book, notes and no electronic devices. When is a prior distribution called dogmatic?

19 Quiz 19. Name: Closed book, notes and no electronic devices. Turn quiz over when done. Bayesian simulation-based methods are popular these days. What values (numbers) do you simulate when using these methods? Include a graph with your answer.

20 Quiz 20. Name: Closed book, notes and no electronic devices. Turn quiz over when done. Suppose Y1, Y2,, Y100 are iid Bernoulli(0.5) outcomes. Thus each Yi has mean = 0.5 and variance 2 = Give the approximate distribution of (Y1 + Y2 + +Y100)/100.

21 Quiz 21. Name: Closed book, notes and no electronic devices. Turn quiz over when done. The stock return example showed that stock returns are not independent of previous returns. State the null model that was used for how the stock return data Y1, Y2,, Y18,834 were produced.

22 Quiz 22. Name: Closed book, notes and no electronic devices. Turn quiz over when done. 1. Draw a graph of the standard normal distribution. Put numbers on the horizontal axis. 2. Draw a graph of the chi-squared distribution with 5 degrees of freedom. Put numbers on the horizontal axis.

23 Quiz 23. Name: Instructions: Closed book, notes and no electronic devices. Turn quiz over when done. The two-sample t test is used to compare means of data in two different groups. The data are Yij, where i denotes group (i = 1,2), and j indicates observation within group (j = 1,2,,ni). State the null model that you assume to produce the data Yij when you use the two-sample t test.

24 Quiz 24. Name: Instructions: Closed book, notes and no electronic devices. Turn quiz over when done. In the quality control example, the null mean was 310. State the null model that was assumed to produce the quality control measurements Y1,, Yn.

25 Quiz 25. Name: Instructions: Closed book, notes and no electronic devices. Turn quiz over when done. In the (George HW Bush, Barbara Bush) example, the unrestricted model was (X1,Y1), (X2,Y2),..., (X33, Y33) ~iid p(x,y). What was different about the restricted (null) model?

26 Quiz 26. Name: Instructions: Closed book, notes and no electronic devices. Turn quiz over when done. Suppose you will collect (X,Y) data like the George/Barbara Bush case to see if X and Y are independent. After collecting the data, you will perform the test for independence of X and Y. What is a Type II error in this case?

27 Quiz 27. Name: Instructions: Closed book, notes and no electronic devices. Turn quiz over when done. 1. Which distribution produces outliers? A. Uniform B. Cauchy C. Normal D. Bernoulli 2. How does one simulate data when performing a permutation test? A. By randomly shuffling the observed data set, like a deck of cards B. By generating iid samples from the same generic distribution C. By generating iid samples from the same normal distribution D. By generating iid samples from the same Cauchy distribution 3. When should you use a bootstrap confidence interval? A. When the distribution that produced the data differs greatly from normal B. When you know the distribution that produced the data C. When the sample size is small D. When the power of the t-test is too low 4. How many bootstrap samples should you take? Pick the best answer. A. 30 B. 100 C. 1,000 D. 20,000