Exam B: Statistical Inference and Hypothesis Testing, Exams of Economic statistics

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Stuff I Need to Know for Econ 15A-B

These questions represent everything you need to know for the upcoming exam (only as far as we have covered in lecture, of course). Bulleted items are important and useful, but they will not be questions on any exams; only the numbered items will be relevant to the exam. ========================================================= Exam B3 things to know Whenever possible, show all your work. If the problem allows you to calculate a particular number, or some expression shown in class, you must calculate that number/expression, and reduce all fractions. Unless specified otherwise, questions about confidence intervals refer to confidence intervals for the mean, with the variance known. Use the T-distribution values in this table whenever they are given; if the appropriate T- distribution is not here, use the normal distribution instead. For this exam only, when reporting the results of a test, you do not need to report the p-value. p = .900 .905 .910 .915 .920 .925 .930 .935 .940 .945. @qnorm(p) = 1.28 1.31 1.34 1.37 1.41 1.44 1.48 1.51 1.55 1.60 1. @tdist(p, 4) = 1.53 1.58 1.62 1.67 1.72 1.78 1.84 1.90 1.97 2.05 2. @tdist(p, 5) = 1.48 1.52 1.56 1.60 1.65 1.70 1.75 1.81 1.87 1.94 2. @tdist(p, 6) = 1.44 1.48 1.52 1.56 1.60 1.65 1.70 1.75 1.81 1.87 1. p = .955 .960 .965 .970 .975 .980 .985 .990. @qnorm(p) = 1.70 1.75 1.81 1.88 1.96 2.05 2.17 2.33 2. @tdist(p, 4) = 2.23 2.33 2.46 2.60 2.78 3.00 3.30 3.75 4. @tdist(p, 5) = 2.10 2.19 2.30 2.42 2.57 2.76 3.00 3.36 4. @tdist(p, 6) = 2.02 2.10 2.20 2.31 2.45 2.61 2.83 3.14 3.

1. What is a Type I error, written in probability theory notation?
2. What is a Type II error, written in probability theory notation?
3. Describe, in ordinary English, what a Type I error is.
4. Describe, in ordinary English, what a Type II error is.
5.  is directly related to one of the two Types of errors; what Type, and how?
6.  is directly related to one of the two Types of errors; what Type, and how?

7. In what fundamental way is the null hypothesis different from the alternative hypothesis?
8. In a one-sided hypothesis test, what role, if any, does H 1 play in the calculations?
9. Suppose a quantile q has been determined for a one-sided hypothesis test where H 1 : . Show how to calculate from the appropriate inequality involving the t statistic and q to an inequality involving the sample mean estimator and the critical boundary.
10. What are the components of a statistical test, as given in lecture and the slides?
11. Run and report on a statistical test for the mean of a N(, ^2 ), with data {2, 1, 4, 7, 3},  = .05, H 0 :  = –1, H 1 :  = 3.
12. Give the general formula for the t-statistic.
13. Give a formula for the p-value in probability theory notation for a one-sided test, where H 1 : .
14. Describe, in ordinary English, what the p-value tells you.
15. In a two-sided hypothesis test, what role, if any, does H 1 play in the calculations?
16. What things are included in a report of a statistical test?
17. Run and report on a statistical test for the mean of N(, ^2 ), with data {6, 1, 0, 7, 3},  = .05, H 0 :  = –1, H 1 :  > –1.
18. If you ran a statistical test in Eviews, where H 0 :  = 60, and H 1 :  < 60, Eviews would report a larger p-value than is in fact correct. Why?
19. If you ran a statistical test in Eviews, where H 0 :  = 60, and H 1 :  > 60, Eviews would report a larger p-value than is in fact correct. Why?
20. Describe how to run a hypothesis test for the mean where H 0 :  = 45 and H 1 :  < 45; i.e., list the basic quantities to be determined, and how to put them together, and what to compare, and how to conclude.
21. Describe how to run a hypothesis test for the mean where H 0 :  = 45 and H 1 :  > 45; i.e., list the basic quantities to be determined, and how to put them together, and what to compare, and how to conclude..
22. Describe how to run a hypothesis test for the mean where H 0 :  = 45 and H 1 :  ≠ 45; i.e., list the basic quantities to be determined, and how to put them together, and what to compare, and how to conclude.
23. What is the difference between a one-sided and a two-sided hypothesis test?
24. For a test of the equality of means, what is H 0?
25. Explain how increasing  does (or doesn’t) change the chances of a Type I error.
26. Explain how increasing  does (or doesn’t) change the chances of a Type II error.
27. Explain how increasing  does (or doesn’t) change the chances of a Type I error.
28. Explain how increasing  does (or doesn’t) change the chances of a Type II error.
29. Explain how increasing n does (or doesn’t) change the chances of a Type I error.
30. Explain how increasing n does (or doesn’t) change the chances of a Type II error.
31. If every other aspect of a statistical test is the same, which has more power, a one- sided or two-sided hypothesis test? Explain. (Assume that all the relevant assumptions behind both tests are correct.)  Consider the probability formulas for a confidence interval, and for a two-sided hypothesis test. How are they alike? How are they different?
32. “A two-sided test is more powerful, because you test both sides of the distribution.” Explain what is wrong with this claim.

33. What is the primary advantage and primary disadvantage of a two-sided test, as compared with a one-sided test?
34. Give the mathematical formula for the estimate of the variance of the mean in a test for equality of means.
35. If there are 30 observations in the first group, and 38 observations in the second group, how many degrees of freedom are there in a test for the equality of means?
36. Suppose you read that in a survey of 20 UCI students, the average number of hours spent online per day is 6.2, with a standard deviation of 1.4, and that in a survey of 30 UCSD students, the average was 4.8, with a standard deviation of 1.7. Use this data to conduct a test for the equality of means at the .05 level.
37. A cranky comment on your blog complains about “kids these days” spending too much time on “frivolous” activities like Twittering, Facebooking, etc. She cites a study that found that persons aged 20 – 25 spend on average about 3.2 hours per day on these activities, and declares that there is a problem because anything more than 1.5 hours per day is too much. You follow the link to this study and find that the poll used 18 responses and had a standard deviation of 1.9. Conduct a one-sided test, at the .05 level, to see if these data supports this critic’s comments.
38. In a given sample of size 45, if the mean and standard deviation are 6 and 3, respectively, what is the t-statistic for the null hypothesis that the population mean is 7?
39. ** [Not required] Do problem 11.1 in the book.
40. Do problem 11.2 in the book.
41. Do problem 11.3 in the book.
42. Do problem 11.4 in the book.
43. Do problem 11.5 in the book.
44. Do problem 11.6 in the book.
45. Do problem 11.7 in the book.
46. Do problem 11.8 in the book.
47. Do problem 11.11 in the book.
48. Do problem 11.13 in the book.
49. Do problem 11.14 in the book.
50. Do problem 11.15 in the book.
51. Do problem 11.16 in the book.
52. We studied four different types of statistical tests; name them.
53. Describe the hypotheses involved in a significance test between two simple hypotheses; i.e., what is general form of H 0 and H 1?
54. Describe the hypotheses involved in a one-sided significance test; i.e., what is general form of H 0 and H 1?
55. Describe the hypotheses involved in a two-sided significance test; i.e., what is general form of H 0 and H 1?
56. Describe the hypotheses involved in a test for the equality of means; i.e., what is general form of H 0 and H 1?
57. In actual research, one of the background assumptions behind statistical testing is frequently violated. What is this assumption, and why is this violation often not a serious concern?

58. Give a “real life” example of when you might want to use a significance test between two simple hypotheses.
59. Give a “real life” example of when you might want to use a one-sided significance test.
60. Give a “real life” example of when you might want to use a two-sided significance test.
61. Give a “real life” example of when you might want to use a test for the equality of means.
62. Which, if either, is larger, the cumulative distribution of T(8) at -3.2 or the cumulative distribution of T(27) at -3.2. Explain your answer.
63. Which, if either, is larger, the cumulative distribution of T(8) at -3.2 or the cumulative distribution of T(8) at 3.2. Explain your answer.
64. Which, if either, is larger, the cumulative distribution of N(0, 1) at -3.2 or the cumulative distribution of T(8) at -3.2. Explain your answer.
65. Give a formula for the p-value in probability theory notation for a two-sided test.
66. Give a formula for the p-value in probability theory notation for a one-sided test, where H 1 : .
67. Suppose x^ = 10, sx = 7.1, nx = 12, and y^ = 14, sy = 3.2, ny = 15. You want to conduct a hypothesis test for the equality of the means. What are the sum of squared deviations from the mean for x and y?
68. Describe how to run a hypothesis test for the equality of means; i.e., list the basic quantities to be determined, and how to put them together, and what to compare, and how to conclude.
69. Why do we typically use a T-distribution to conduct statistical tests rather than a Normal distribution?
70. Give the specific formula for the t-statistic for a test for equality of means.
71. When conducting a two-sided hypothesis test, why do we divide s by n?
72. When conducting a two-sided hypothesis test, why do we subtract H0 from x , and then divide the result by sx^?
73. If the p-value of a test is .04, is the result significant at  = .05? Why or why not?
74. If the p-value of a test is .06, is the result significant at  = .05? Why or why not?
75. “If we wanted to be really certain of a statistical test, then we would set  = .000000000001.” Explain why this is typically unwise practice.
76. When conducting a two-sided test, why do we use  instead of ?
77. When conducting a two-sided test, why do we double the p-value that we would get if our test was one-sided?
78. Run and report on a statistical test for the mean of N(, ^2 ), with data {6, 1, 0, 7, 3},  = .05, H 0 :  = –1, H 1 :  ≠ –1.
79. Run and report on a statistical test for the equality of means, with data x = {6, 1, 0, 7, 3}, y = {4, 1, 5, 0},  = .01.
80. What are the null and alternative hypotheses for a test of the equality of means?
81. Run and report on a statistical test for the mean where  = .01, H 0 :  = 17, H 1 :  = 22, x^ = 19.7, s = 9.1, n = 24.
82. Run and report on a statistical test for the mean where  = .01, H 0 :  = 17, H 1 :  < 17, x^ = 15.4, s = 9.1, n = 28.

83. Run and report on a statistical test for the mean where  = .03, H 0 :  = 17, H 1 :  > 17, x^ = 19.7, s = 4.1, n = 28.
84. Run and report on a statistical test for the mean where  = .03, H 0 :  = 17, H 1 :  ≠ 17, x^ = 19.7, s = 4.1, n = 28.
85. Run and report on a statistical test for the equality of means where  = .05, H 0 : x = y, H 1 : x ≠ y,, x = 19.7, s = 4.1, nx = 38, y^ = 19.1, s = 2.9, ny = 49. ============================================================== Exam B2 things to know p = .900 .905 .910 .915 .920 .925 .930 .935 .940 .945. @qnorm(p) = 1.28 1.31 1.34 1.37 1.41 1.44 1.48 1.51 1.55 1.60 1. @tdist(p, 4) = 1.53 1.58 1.62 1.67 1.72 1.78 1.84 1.90 1.97 2.05 2. @tdist(p, 12) = 1.36 1.39 1.42 1.46 1.50 1.54 1.58 1.63 1.67 1.73 1. p = .955 .960 .965 .970 .975 .980 .985 .990. @qnorm(p) = 1.70 1.75 1.81 1.88 1.96 2.05 2.17 2.33 2. @tdist(p, 4) = 2.23 2.33 2.46 2.60 2.78 3.00 3.30 3.75 4. @tdist(p, 12) = 1.84 1.91 1.99 2.08 2.18 2.30 2.46 2.68 3. Unless specified otherwise, questions about confidence intervals refer to confidence intervals for the mean, with the variance known. Chap 10
86. Give the definition of a consistent estimator.
87. What is the formula for the Law of Large Numbers?
88. In words, what does the Law of Large Numbers say?
89. Explain why the Law of Large Numbers is true
90. Give the definition of a biased estimator.
91. Show that X is an unbiased estimator of .

7. Is ^ 





n i X i X n (^) 1

an unbiased estimator of ^2? If not, give its bias.  Give the definition of the mean squared error of an estimator.  Use the definition of the MSE of an estimator to calculate that the MSE is equal to the estimator’s variance plus its squared bias.

8. Give the formula for S^2. Why do we use this estimator?
9. What is a point estimate?
10. What information does a confidence interval supply that point estimates do not?
11. Why are confidence intervals useful?
12. What are the two reasons why we often only need the N(0, 1) distribution when we constructed confidence intervals for X , when the variance was known?
13. Describe what the quantile function does (i.e, it takes you from what to what?).
14. What is the relation between the quantile function and the cdf?

15. Suppose X ~ N(0, 1). Explain how we use the quantile function to translate our choice of a confidence level into a confidence interval for X.
16. What is the formula for the lower bound of a confidence interval when the variance is known? Label the parts.
17. What is the formula for the upper bound of a confidence interval when the variance is known? Label the parts.
18. What is the formula for the lower bound of a confidence interval when the variance is unknown? Label the parts.
19. What is the formula for the upper bound of a confidence interval when the variance is unknown? Label the parts.
20. When the variance is known, what distribution do we refer to?
21. When the variance is unknown, what distribution do we refer to?
22. Practically speaking, what does the t-distribution do that the Normal does not?
23. From which distribution, N(0,1) or T(8), are you more likely to produce a number larger than 3? Explain why.
24. What are the boundaries of a confidence interval at the .99 confidence level if the variance in the 14 samples is known to be 6, and the sample mean was 8?
25. The firm you work for has a rather poorly-thought-out policy that any estimations of the cost of a certain activity must be within a margin of error (interpreted as a confidence interval at some confidence level.) of no more than ± 10% of the estimated cost. A project that you’re working on has an average cost of $10,000, which was derived from a sample of all 11 known observations with a known standard deviation of 1,869. To comply with your company’s policy, what is the highest confidence level you may use? (Estimate this by using the chart above to give the two points which it must fall in between, or the extreme point if it is greater/less than all those given.)
26. You work in an IT firm that specializes in a certain sort of information retrieval for corporations. Your boss has assigned you the task of estimating how long a certain sort of advanced retrieval takes on average. Since a major business is seriously considering an account with your company, your boss wants you to be “99 percent sure” that your estimate falls within a 10-point total spread that you will give him. The variance of these sorts of retrievals is known to be 4800. What is the least number of trials of the system that you would have to run to complete your task (using the techniques that this test covers)?
27. Your colleague at work was gathering some data for you, and you noticed that she accidentally estimated the standard deviation as the square root of the average of the sum of the squared deviations from the sample mean. Her estimate was 270 for the 15 observations she collected. You wish to use an unbiased estimate of the sample variance. What is it (show your work)?
28. Calculate a 99% confidence interval for a data set of 14 observations, whose mean is 6.2, and whose variance is known to be 7.
29. Calculate a 99% confidence interval for a data set of 14 observations, whose mean is 6.2, and whose standard deviation is known to be 7.
30. When constructing confidence intervals using the normal distribution, the book discusses two reasons for choosing a lower bound that is the negative value of the upper bound. What are these reasons?

31. If you are using student’s T distribution with 11 degrees of freedom, how large was your sample?
32. Calculate a 99% confidence interval for a data set of 14 observations, whose mean is 6.2, and whose variance is estimated to be 7.
33. Calculate a 99% confidence interval for a data set of 14 observations, whose mean is 6.2, and whose standard deviation is estimated to be 7.
34. Calculate a 95% confidence interval for the data set {3, 5, 2, 1, 3}. Treat your estimate of the variance as if it were known to be the true variance.
35. Calculate a 95% confidence interval for the data set {3, 5, 2, 1, 3}. Recognize that your estimate of the variance is only an estimate.
36. For a given data set, which is larger, the length of a 95% confidence interval, or the length of a 99% confidence interval? Explain your answer.
37. Suppose two data sets x and y have identical means and variances, but x is made up of 17 observations, and y contains 20. Which will have a larger confidence interval? Explain your answer.
38. For a given data set you construct 99% confidence intervals using first the standard normal distribution, and then using the appropriate t-distribution. Which is larger, and why?
39. Draw the graph of the quantile function for N(0,1); include a numeric scale for both axes, and label the points on the graph for the quantiles corresponding to the probabilities: .005, .025, .05, .95, .975, and .995.
40. If you were constructing a confidence interval for a collection of 17 data points (where the variance is unknown), what distribution exactly would you use?
41. If you were constructing a 99% confidence interval for a collection of 17 data points (where the variance is unknown), what quantile would you have Eviews calculate for you? (Don’t just give the command; say in English what you would be asking Eviews to calculate.)
42. How many times larger will your margin of error be if you switch from a 95% to 99% confidence level (show your work)?
43. How many times larger will the length of your confidence interval be if you switch from a 90% to a 95% confidence level (show your work)?
44. If your sample size is n, how much larger must you make your sample size if you want your margin of error to be cut in half? (Assume all other quantities remain fixed.)
45. Since the quantile function is related to the cdf, it is a little unclear how it can be used to determine the upper quantile. Explain how this works.
46. Suppose you constructed 200 95% confidence intervals for the mean from one population, using a sample of size 100. What is the expected number of intervals that would contain ? What is the expected number that would not?  What is the formula for a confidence interval for a proportion?  In a recent study of 3,000 Crackle.com visitors, 2,016 visitors watched Starship Troopers. Construct a 98% confidence interval for the true proportion of Crackle’s Starship Troopers viewers.  Give two examples of where there would be great practical benefit in estimating a proportion and calculating a confidence interval, and explain why your examples are good ones.

47. Give two examples of where there would be great practical benefit in estimating a mean (not a proportion) and calculating a confidence interval, and explain why your examples are good ones.
48. Give an example of where there would be great practical benefit in estimating a mean (not a proportion) and calculating a confidence interval using a T-distribution, and explain why your examples are good ones.
49. Describe a study (real or fictional) where there are at least three sources of error or uncertainty that are not accounted for by our methods for constructing confidence intervals.
50. Sketch the graphs of the pdfs of N(0, 1) and T(5) on one set of axes. Label each graph, and numerically label the two axes.
51. Give the formula for the margin of error (when the variance is known).
52. Give the formula for the margin of error (when the variance is unknown).
53. What happens to the length of a confidence interval if all quantities remain the same except that x is doubled? Calculate/explain your answer.
54. What happens to the size of the margin of error if all quantities remain the same except that x^ is doubled? Calculate/explain your answer.
55. Your 94% margin of error had a length of 7. What is the size of the margin of error if you change the confidence level to 98%?
56. Your 94% confidence interval had a length of 7. What is the length of the confidence interval if you change the confidence level to 98%?
57. Do exercise 10.3 in the book.
58. Do exercise 10.7 in the book.
59. Do exercise 10.10 in the book.
60. Do exercise 10.11 in the book.
61. Do exercise 10.12 in the book.
62. Do exercise 10.13 in the book.
63. Do exercise 10.14 in the book.
64. Do exercise 10.15 in the book.
65. Do exercise 10.16 in the book.
66. Do exercise 10.17 in the book.
67. Do exercise 10.22 in the book.
68. Do exercise 10.23 in the book.
69. Do exercise 10.24 in the book.
70. The firm you work for has a rather poorly-thought-out policy that any estimations of the cost of a certain activity must be within a margin of error (interpreted as a confidence interval at some confidence level.) of no more than ± 15% of the estimated cost. A project that you’re working on has an average cost of $15,000, which was derived from a sample of all 13 known observations where the standard deviation was estimated to be 5,668. To comply with your company’s policy, what is the highest confidence level you may use? (Estimate this by using the chart above to give the two points which it must fall in between, or the extreme point if it is greater/less than all those given.)
71. Suppose you have decided to gather some fixed number n of observations, and create a confidence interval at some confidence level c, where the variance of the underlying

population is known to be ^2. Before you even gather your n numbers, you already know how long the confidence interval will be. Explain why.

72. Suppose you have decided to gather some fixed number n of observations, and create a confidence interval at some confidence level c, where the variance of the underlying population will be estimated. Before you gather your n numbers, don’t know how long your confidence interval will be. Explain why.
73. Suppose you have decided to gather some fixed number n of observations, and create a confidence interval at some confidence level c, where the variance of the underlying population is known to be ^2. Before you even gather your n numbers, you already know how big your margin of error will be. Explain why.
74. Suppose you have decided to gather some fixed number n of observations, and create a confidence interval at some confidence level c, where the variance of the underlying population will be estimated. Before you gather your n numbers, don’t know how big your margin of error will be. Explain why.
75. If you have two data sets that yield identical estimates of the variance, but for the first one you used 20 observations and for the second you used 30, which one will have the larger margin of error? Why?
76. Describe in words what a consistent estimator is.
77. Give the mathematical formula that characterizes a consistent estimator.
78. Describe in words what a biased estimator is.
79. Describe in words what an unbiased estimator is.
80. Give the mathematical formula that characterizes an unbiased estimator.
81. Describe the role that the Central Limit Theorem plays in our construction of confidence intervals.
82. Give the Eviews command for the quantile appropriate for a 98% confidence interval using 34 observations, when the variance is only estimated.
83. Give the Eviews command for the quantile appropriate for a 98% confidence interval using 34 observations, when the variance is known. ========================================================= Exam B1 things to know Chap 08
84. What do pdf and pmf stand for? What kinds of distributions do they respectively apply to?
85. What conditions must a function f meet in order to be a pdf?
86. For a given distribution, give the formula for expressing the cdf in terms of the pdf. Write a sentence about what this formula means.
87. For a given distribution, give the formula for expressing the pdf in terms of the cdf. Write a sentence about what this formula means.
88. Suppose X has a continuous distribution. What is pr(X = 0)? Explain your answer.
89. Express pr(X < 6.2) in terms of the cdf F.
90. Express pr(X > 6.2) in terms of the cdf F.
91. Express pr(5.1 < X < 6.2) in terms of the cdf F.

9. Express pr(X < 6.2) in terms of the pdf f.
10. Express pr(X > 6.2) in terms of the pdf f.
11. Express pr(5.1 < X < 6.2) in terms of the pdf f.
12. Draw the and label (both axes of) the graphs of the pdf (together, on one set of axes) of U(0,1) and U(-1, 2).
13. Draw and label (both axes of) the graphs of the cdf (together, on one set of axes) of U(0,1) and U(-1, 2).
14. What is the pdf for the Uniform distribution on (a, b)?
15. What is the cdf for the Uniform distribution on (a, b) (do not express this using integrals)?
16. What is the pdf for N(, ^2 )?
17. Draw and label the graphs of the pdf (together, on one set of axes) of N(0,1) and N(- 1, 2).
18. Draw and label the graphs of the cdf (together, on one set of axes) of N(0,1) and N(-1, 2).
19. Explain what the Central Limit Theorem says. Be sure to include the background conditions necessary for the theorem to hold, and the “limit statement” as presented in the book that gives the mathematical essence of the theorem.
20. Give the limit formula in the Central Limit Theorem, not as it is presented in the book, but as a limit formula of a standardization of a random variable. In this formulation, what random variable is being standardized?
21. If X ~ U(2, 7), calculate pr(X ≤ 4) using only the definitions of U(2, 7) and pr(X ≤ 4).
22. If X ~ U(2, 7), calculate pr(X ≥ 4) using only the definitions of U(2, 7) and pr(X ≥ 4).
23. If X ~ U(2, 7), calculate pr(3 ≤ X ≤ 4) using only the definitions of U(2, 7) and pr(3 ≤ X ≤ 4).
24. Starting from the definitions of a uniform distribution and expectation, calculate E[X], if X ~ U(2, 7).
25. Starting from the definition of a uniform distribution, calculate E[X] to the form we saw in class, if X ~ U(a, b).  Starting from the definition of a uniform distribution, calculate the variance of X to the form we saw in class, if X ~ U(a, b).
26. What command would you give if you wanted Eviews to calculate the probability that X < .2, if X ~ U(0, 1)?
27. Suppose Y is the standardization of X (Y =  X   , where  and  are the mean and standard deviation of X). Calculate the mean of Y.
28. Suppose Y is the standardization of X (Y =  X   , where  and  are the mean and standard deviation of X). Calculate the standard deviation of Y.
29. If X is a continuous probability distribution, why doesn’t it matter whether you determine pr(X < 5.1) or pr(X ≤ 5.1)?
30. What is the standardized skew of N(, ^2 )?
31. What is the standardized kurtosis of N(, ^2 )?
32. How much of the probability of an N(, ^2 ) distribution is within 1.96 standard deviations of the mean?
33. If X ~N(, ^2 ), what is what is the probability that X ≤ ( + 1.64)?

34. Why does the normal distribution occur frequently in nature?
35. Give three examples of populations which are probably (nearly) normally distributed; for each one, write a sentence or two about why they probably are normally distributed.
36. For N(, ^2 ), calculate the largest value of its pdf. Show also that any other value will be less than this number.
37. What command would you use if you wanted Eviews to compute the probability that X < 3, if X~N(0, 1)?
38. Do exercises (1) – (4) 8.3a in the book.
39. Do exercise 8.4 in the book.
40. Do exercise 8.5 in the book.
41. Do exercise 8.6 in the book.
42. Do exercise 8.10 in the book.
43. If X ~ N(0, 1), then what is the distribution of Y = a + bX (b ≠ 0)?
44. If X ~ N(, ^2 ), how would you transform it so that it is distributed as N(0, 1)?
45. If X ~ N(, ^2 ), then what is the distribution of Y = a + bX (b ≠ 0)?
46. If X ~ N(0, 1), then what is the distribution of Y = 2 + 3X?
47. If X ~ N(, ), how would you transform it so that it is distributed as N(0, 1)?
48. If X ~ N(, ), then what is the distribution of Y = 2 + 3X?
49. For what values is the pdf of N(6, 4) greater than 0?
50. Write a couple sentences explaining the differences between x , (^) X , and 
51. Calculate the value of E^ (^ X ) in terms of the mean  of Xi, (i.e., like we did in class).
52. Calculate the variance of (^) X in terms of the variance ^2 of Xi (i.e., like we did in class).
53. If {X 1 ,…, X 25 } is a random sample from N(100, 25), then what is the variance of (^) X?
54. If {X 1 ,…, X 25 } is a random sample from N(100, 25), then what is the standard deviation of X?
55. Suppose X ~ U(0, 1). Which, if either, is more likely: .3 ≤ X ≤ .4, or .7 ≤ X ≤ .8? Explain your answer.
56. Suppose X ~ U(0, 1). Which, if either, is more likely: .3 < X < .4, or .7 ≤ X ≤ .8? Explain your answer.
57. Suppose X ~ U(0, 1). Which, if either, is more likely: .3 ≤ X ≤ .41, or .7 ≤ X ≤ .8? Explain your answer.
58. Suppose X ~ N(0, 1). Which, if either, is more likely: .3 ≤ X ≤ .4, or .7 ≤ X ≤ .8? Explain your answer.
59. Suppose X ~ N(0, 1). Which, if either, is more likely: .3 < X < .4, or .7 ≤ X ≤ .8? Explain your answer.
60. Suppose X ~ N(0, 1). Which, if either, is more likely: –1 ≤ X ≤ 1, or 5 ≤ X ≤ 8? Explain your answer.
61. What is the total area under the curve of the pdf of N(0, 1)?
62. What is the total area under the curve of the pdf of U(2, 4)?
63. What is E[X] if X ~ U(a, b)?
64. What is E[(X – E[X])^2 ] if X ~ U(a, b)?
65. What is E[X] if X ~ U(2, 7)?
66. What is E[(X – E[X])^2 ] if X ~ U(2, 7)?
67. What is E[X] if X ~ N(, ^2 )?

68. What is E[(X – E[X])^2 ] if X ~ N(, )?
69. What is E[X] if X ~ N(, ^2 )?
70. What is E[(X – E[X])^2 ] if X ~ N(, )?
71. The book gives two names for N(, ^2 ). What are they? Chap 09   Do exercise 9.2 in the book.  Do exercise 9.4 in the book.  Do exercise 9.13 in the book.  Do exercise 9.16 in the book.  Do exercise 9.17 in the book. ========================================================= 15 A Stuff Chapter 02
72. What kind of thing is a random variable?
73. Know what the four scales of measurement that we cover are (ratio, interval, ordinal, and categorical), and how to describe them. I.e., what is distinctive about each?
74. How are the four scales of measurement that we cover related to one another (i.e., is one sort of scale always an instance of another sort of scale)?
75. Give two examples of something that would be measured on a ratio scale, and write a sentence or two about why these are good examples of this scale.
76. Give two examples of something that would be measured on an interval scale, and write a sentence or two about why these are good examples of this scale.
77. Give two examples of something that would be measured on an ordinal scale, and write a sentence or two about why these are good examples of this scale.
78. Give two examples of something that would be measured on a categorical scale, and write a sentence or two about why these are good examples of this scale.
79. Do the same for indexical and time series scales. Chapter 03
80. How would you determine the median of a data set?
81. When is the median an element of a data set? When is it not?
82. Describe how the box of a box-and-whisker plot is constructed.
83. How far out do the whiskers extend?
84. What are the outliers (if any) in a box-and-whisker plot?
85. Know how to determine the three quartile points and the four quartiles of a box-and- whisker plot
86. Know how to construct a box-and-whiskers plot (by hand) for a relatively small data set, e.g., {3, 4, 3, 2, 1, -8, 4, 3, 2, 0}.
87. What kind of information does a box-and-whisker plot visually display?
88. Give an example of when and how you might use a box-and-whisker plot.
89. How are relative frequency plots different from cumulative frequency plots?

19. Know how to sketch a relative frequency plot and a cumulative distribution plot (by hand) for a relatively small data set, e.g., 2% of students are under 18, 10% are 18, 20% are 19, 25% are 20, 20% are 21, 15% are 22, and 8% are over 22.
20. What kind of information does a relative frequency plot visually display?
21. Give an example of when and how you might use a relative frequency plot.
22. Know how to sketch a cumulative distribution plot (by hand) for a relatively small data set, e.g., 2% of students are under 18, 10% are 18, 20% are 19, 25% are 20, 20% are 21, 15% are 22, and 8% are over 22.
23. What kind of information does a cumulative distribution plot visually display?
24. Give an example of when you might use a cumulative distribution plot.
25. How are histograms constructed?
26. What is the total area of a histogram?
27. Know how to sketch a histogram (by hand) for a relatively small data set, e.g., {1, 1, 3, 2, 5, 5, 9, 2, 4, 2}
28. What kind of information does a histogram visually display?
29. Give an example of when you might use a histogram.
30. How do histograms and relative frequency plots differ? What different kinds of information do they display? Chapter 04
31. For m’ 1 , give: Its name
32. For the mean, give: Its symbol or symbols, if there is more than one
33. For m’ 1 , give: Its mathematical formula (using sigma notation)
34. For m’ 1 , give: A one sentence characterization of what it measures
35. For m’ 1 , give: A brief description of how you could make it larger by altering the data set.
36. For m’ 1 , give: A brief description of how you could make it smaller by altering the data set.
37. For m’ 1 , give: Using a calculator, how to calculate its value, in the way we do it in class (i.e., showing your work at each step), for a small data set, e.g., {3, 4, -5, 8, 4}, {12, 3, -7, 2}
38. For m’ 1 , give: At least two “real life” examples where knowing that moment would provide you with valuable information. (Be sure to be able to explain why your examples are good ones.)
39. When data (e.g., household incomes, annual profits) are grouped into categories, what is the cell mark of each category?
40. For m 2 , give: Its name
41. For the variance, give: Its symbol or symbols, if there is more than one
42. For m 2 , give: Its mathematical formula (using sigma notation)
43. For m 2 , give: A one sentence characterization of what it measures
44. For m 2 , give: A brief description of how you could make it larger by altering the data set.
45. For m 2 , give: A brief description of how you could make it smaller by altering the data set.
46. What does a large value of m 2 indicate about a data set?
47. What does a small value of m 2 indicate about a data set?

48. For m 2 , give: Using a calculator, how to calculate its value, in the way we do it in class (i.e., showing your work at each step), for a small data set, e.g., {-3, 4, 5, 8, 4}, {12, 3, 7, -2}
49. For m 2 , give: At least two “real life” examples where knowing that moment would provide you with valuable information. (Be sure to be able to explain why your examples are good ones.)
50. For m 2^ , give: Its name
51. For the standard deviation, give: Its symbol or symbols, if there is more than one
52. For m 2^ , give: Its mathematical formula (using sigma notation)
53. For m 2^ , give: A one sentence characterization of what it measures
54. For m 2^ , give: A brief description of how you could make it larger by altering the data set.
55. For m 2 , give: A brief description of how you could make it smaller by altering the data set.
56. For m 2^ , give: Using a calculator, how to calculate its value, in the way we do it in class (i.e., showing your work at each step), for a small data set, e.g., {3, -4, 5, 8, 4}, {12, 3, -7, 2}
57. For m 2^ , give: At least two “real life” examples where knowing that moment would provide you with valuable information. (Be sure to be able to explain why your examples are good ones.)
58. For m 3 , give: Its name
59. For the unstandardized skew, give: Its symbol or symbols, if there is more than one
60. For m 3 , give: Its mathematical formula (using sigma notation)
61. For m 3 , give: A one sentence characterization of what it measures
62. For m 3 , give: A brief description of how you could make it larger by altering the data set.
63. For m 3 , give: A brief description of how you could make it smaller by altering the data set.
64. What does a large value of m 3 indicate about a data set?
65. What does a small value of m 3 indicate about a data set?
66. For m 3 , give: Using a calculator, how to calculate its value, in the way we do it in class (i.e., showing your work at each step), for a small data set, e.g., {3, 4, -5, 8, 4}, {12, 3, -7, 2}
67. For m 3 , give: At least two “real life” examples where knowing that moment would provide you with valuable information. (Be sure to be able to explain why your examples are good ones.)
68. Why do we often standardize m 3?
69. For m 4 , give: Its name
70. For the unstandardized kurtosis, give: Its symbol or symbols, if there is more than one
71. For m 4 , give: Its mathematical formula (using sigma notation)
72. For m 4 , give: A one sentence characterization of what it measures
73. For m 4 , give: A brief description of how you could make it larger by altering the data set.
74. For m 4 , give: A brief description of how you could make it smaller by altering the data set.
75. What does a large value of m 4 indicate about a data set?
76. What does a small value of m 4 indicate about a data set?

77. For m 4 , give: Using a calculator, how to calculate its value, in the way we do it in class (i.e., showing your work at each step), for a small data set, e.g., {3, 4, -5, 8, 4}, {12, 3, -7, 2}
78. For m 4 , give: At least two “real life” examples where knowing that moment would provide you with valuable information. (Be sure to be able to explain why your examples are good ones.)
79. Why do we often standardize m 4?
80. For ^ ˆ 1 , give: Its name
81. For the standardized skew, give: Its symbol or symbols, if there is more than one
82. For ^ ˆ 1 , give: Its mathematical formula (using sigma notation)
83. For ^ ˆ 1 , give: A one sentence characterization of what it measures
84. For ^ ˆ 1 , give: A brief description of how you could make it larger by altering the data set.
85. For ^ ˆ 1 , give: A brief description of how you could make it smaller by altering the data set.
86. For ^ ˆ 1 , give: Using a calculator, how to calculate its value, in the way we do it in class (i.e., showing your work at each step), for a small data set, e.g., {3, 4, 5, 8, 4}, {12, 3, 7, 2}
87. For ^ ˆ 1 , give: At least two “real life” examples where knowing that moment would provide you with valuable information. (Be sure to be able to explain why your examples are good ones.)
88. For ^ ˆ^2 , give: Its name
89. For the standardized kurtosis, give: Its symbol or symbols, if there is more than one
90. For ^ ˆ^2 , give: Its mathematical formula (using sigma notation)
91. For ^ ˆ^2 , give: A one sentence characterization of what it measures
92. For ^ ˆ^2 , give: A brief description of how you could make it larger by altering the data set.
93. For ^ ˆ^2 , give: A brief description of how you could make it smaller by altering the data set.
94. For ^ ˆ^2 , give: Using a calculator, how to calculate its value, in the way we do it in class (i.e., showing your work at each step), for a small data set, e.g., {3, 4, 5, 8, 4}, {12, 3, 7, 2}
95. For ^ ˆ^2 , give: At least two “real life” examples where knowing that moment would provide you with valuable information. (Be sure to be able to explain why your examples are good ones.)
96. How do you construct standardized scores?
97. Know how to calculate the mean of a data set whose entries have been standardized.
98. Know how to calculate the variance (here, m 2 ) of a data set whose entries have been standardized.
99. What is the relation between ^ ˆ 1 and m 3?
100. What is the relation between ^ ˆ^2 and m 4? Eviews
101. Describe how you would use Eviews to construct a histogram.

102. Describe how you would use Eviews to construct a box-and-whiskers plot.
103. Describe how you would use Eviews to construct a cumulative distribution plot.
104. Describe how you would use Eviews to construct a scatterplot.
105. Describe how you would use Eviews to construct a correlation matrix between 3 series.
106. Describe how you would use Eviews to construct a covariance matrix between 3 series.
107. What would you write the white syntax bar if you wanted to restrict your sample to those observations for which the variable (series) x was greater than or equal to 6 but less than 12?
108. What would you write in the white syntax bar if you wanted to create a new variable y that was related to an existing variable (series) by: y = 200 - .1x?
109. What would you write in the white syntax bar if you wanted to know the mean of x?
110. What would you write in the white syntax bar if you wanted to know the standard deviation of x? Chapter 05
111. Calculate the covariance of the data set {(1, 0), (0, 1), (2, -1), (3, 1)}.
112. Calculate the correlation of the data set {(1, 0), (3, 1), (0, -1), (1, 1)}.
113. Standardize the data set {(1, 0), (3, 1), (0, -1), (1, 1)} and then calculate the correlation.
114. If the standard deviation of x and y are 4 and 7 respectively, then what is the correlation between x and y if m 11 = 12?
115. What are the possible values of the covariance?
116. What are the possible values of r?
117. What does it mean if r is large and positive?
118. What does it mean if r is nearly zero?
119. What does it mean if r is a large negative number (i.e., not close to zero)?
120. What is the symbol for the correlation coefficient?
121. Express the correlation coefficient in terms of moments.
122. What is the symbol for the covariance?
123. Using sigma notation, give the mathematical formula for the correlation coefficient.
124. Using sigma notation, give the mathematical formula for the covariance.
125. What precisely does the correlation coefficient measure?
126. Give two examples where you would want to know the correlation between two variables.
127. Explain the relationship between covariance and correlation in terms of standardized variables.
128. Explain the relationship between covariance and correlation in terms of moments.
129. Give an example of why “correlation does not imply causation”.
130. Why can’t you take a sample of Californians and a sample of New Yorkers, and measure the correlation between the two groups’ salaries?
131. Know how to do exercise 5.12 in the book.
132. Know how to do exercise 5.13 in the book.

133. Know how to do exercise 5.20 in the book.
134. Know how to do exercise 5.22 in the book.
135. Know how to do exercise 5.29 in the book. X=1 X = 2 Y = 1 5 2 Y = 2 4 1 Y = 3 1 7
136. Calculate the joint probability that both X and Y are 2.
137. Calculate the marginal probability that Y = 3.
138. Calculate the probability that Y = 1 given that X = 2. X=1 X = 2 Y = 1 5 2 Y = 2 or 3 5 8
139. Calculate the correlation between X and Y.
140. Give the mathematical formula for^ ˆ^. Chapter 06
141. What is the definition of a sample space?
142. In a given trial or experiment, how many of the possible outcomes in the sample space can occur? How many must occur?
143. What are the three axioms of probability?
144. What is the general formula for the probability of event A or B occurring?
145. What is the formula for the conditional probability of event A occurring, given that event B occurs?
146. Give the definition of statistical independence of two events.
147. Give the definition of statistical independence of two random variables.
148. Give the definition of statistical independence of n random variables.
149. Prove that if both A and B have non-zero probabilities, and if A is statistically independent of B, then B is statistically independent of A.
150. What does i.i.d. stand for?
151. Which, if any, of the purported assignment of probabilities to the outcomes in a sample space are legitimate? Why or why not? Table 1 Outcomes in the sample space E1 E2 E3 E4 E5 E6 E7 E pr 1 .2 .2 .1 .05 .05 .1 .3 0 pr 2 .3 .1 .1 .2 .1 .1 .1. pr 3 .3 .05 .04 .05 .1 .1 .1. pr 4 .2 .2 .3 .1 –.1 .1 .1. Table 2 Outcomes in the sample space E1 E2 E3 E4 E5 E6 E pr .2 .3 .1 .05 .05 .1. A = {E1, E2}, B = {E2, E3, E4}, C = {E6}, D = {E2, E7, E6}, F = {E1, E7}
152. Using Table 2, show your work as you calculate pr(A or B) and pr(B or C).
153. Using Table 2, show your work as you calculate pr(A and B) and pr(B and C).

154. Using Table 2, show your work as you calculate pr(A|B) and pr(B|C).
155. Using Table 2, show your work as you determine whether A and D are independent.
156. Using Table 2, show your work as you determine whether A and F are independent.
157. Do exercise 6.3 in the book.
158. Do exercise 6.6 in the book.
159. Do exercise 6.7 in the book.
160. Do exercise 6.8 in the book.
161. Do exercise 6.9 in the book.
162. Do exercise 6.18 in the book.
163. Do exercise 6.25 in the book.
164. Do exercise 6.26 in the book.
165. Do exercise 6.27 in the book.
166. Do exercise 6.28 in the book.
167. Construct a numerical example that shows a case where pr(A) < pr(B|A).
168. Construct a numerical example that shows a case where pr(A) = pr(B|A).
169. Construct a numerical example that shows a case where pr(A) > pr(B|A).
170. Draw a diagram that displays a case where pr(A) < pr(A | B).
171. Draw a diagram that displays a case where pr(A) = pr(A | B).
172. Draw a diagram that displays a case where pr(A) > pr(A | B).
173. Construct a numerical example that shows a case where pr(A|B) < pr(B|A).
174. Construct a numerical example that shows a case where pr(A|B) = pr(B|A).
175. Construct a numerical example that shows a case where pr(A|B) > pr(B|A).
176. Draw a diagram that displays a case where pr(A|B) < pr(B|A).
177. Draw a diagram that displays a case where pr(A|B) = pr(B|A).
178. Draw a diagram that displays a case where pr(A|B) > pr(B|A).
179. Write a sentence describing what Simpson’s paradox is (do not just give an example).
180. Give a numerical example of Simpson’s paradox, and calculate the relevant values to show that it is an example.
181. Explain the conditions that make Simpson’s paradox occur.
182. Do exercise 6.36 in the book. Chapter 07
183. Be able to calculate n! e.g., what is 12!?
184. How many ways are there to order 10 of 10 objects?
185. How many ways are there to order k of n objects (k < n)?
186. How many ways are there to order 4 of 10 objects?
187. How many ways are there to pick out (unordered) 6 objects from a group of 12?
188. Give the mathematical formula for (^)       k n .
189. Calculate (^)       4 9 by hand (do the calculations explicitly).
190. What does (^)       k n express?

191. Explain why, or simply calculate that: (^)        ^       n k n k n .
192. The next few questions refer to this set of (actual) students. {Ryan, Enrique, Mallory, Muhammad, Arsen, Kristine, Tevan, Adir, Mohammed}
193. How many students are there?
194. How many ways can these students be ordered?
195. How many groups of 3 students are there?
196. How many groups of 5 students are there that contain Arsen and Tevan?
197. What is the probability of randomly selecting 4 students, and getting Muhammad and Mohammed in your group?
198. Suppose you identify each student by the first letter of their first name (i.e., you don’t distinguish students whose first name starts with the same first letter). How many ways can these students be ordered now?
199. There is a class of 20 with 13 boys and 7 girls. How many ways are there to form a committee of 3 boys and 3 girls?
200. There is a class of 20 with 13 boys and 7 girls. A committee will be formed by randomly selecting 6 persons. What is the probability that the committee will have 3 boys and 3 girls?
201. Give a real-life example where you would want to use the binomial coefficient (just to find a number, not to calculate a probability).
202. Give a real-life example where you would want to calculate a probability using binomial coefficients (not in the context of a binomial distribution).
203. Give the mathematical formula for the binomial distribution.
204. In the formula for the binomial distribution, what are the possible values for p?
205. In the formula for the binomial distribution, what are the possible values for n?
206. In the formula for the binomial distribution, what are the possible values for k?
207. Give two real-life examples where you would want to use the binomial distribution to calculate a probability.
208. What are the three background assumptions regarding the underlying sample for the binomial distribution?
209. What is the mathematical formula for the cumulative distribution function of the binomial distribution?
210. What is the symbol for the mean of a probability distribution?
211. What is the symbol for the variance of a probability distribution?
212. What is the formula for the mean of a probability distribution?
213. What is the formula for the variance of a probability distribution?
214. Suppose pr(X = 1) = p, and pr(X = 0) = (1 – p). Calculate the mean of X.
215. Suppose pr(X = 1) = p, and pr(X = 0) = (1 – p). Calculate the variance of X.
216. For any given person, there is a 25% chance of having a certain inherited genetic trait. There are five unrelated persons in a room. Use the formula for the binomial distribution to calculate the probability that three of them have the trait.
217. For any given person, there is a 25% chance of having a certain inherited genetic trait. There are five siblings in a room. Explain why you can’t use the binomial distribution to calculate the probability that three of them have the trait.
218. What is the mean of the binomial distribution with parameters n and p?
219. What is the mean of the binomial distribution with parameters 4 and .8?

220. What is the variance of the binomial distribution with parameters n and p?
221. What is the variance of the binomial distribution with parameters 4 and .8?
222. Do exercise 7.3 in the book.
223. Do exercise 7.4 in the book.
224. Do exercise 7.9 in the book.
225. Do exercise 7.10 in the book.
226. Do exercise 7.18 in the book (assume all trials are independent).
227. Do exercise 7.21 in the book
228. Do exercise 7.22 in the book.
229. Do exercise 7.23 in the book.
230. What would you type into the white syntax bar on Eviews if you wanted to know the probability of getting 12 successes from a binomial distribution with parameters n = 30 and p = .47?
231. What would you type into the white syntax bar on Eviews if you wanted to know the cumulative probability for 12 successes from a binomial distribution with parameters n = 30 and p = .47?
232. Explain the difference between a probability mass function and a cumulative distribution function.
233. Give the definition of the expectation of a (discrete) random variable X.
234. Give the mathematical formula for the mean of a (discrete) random variable X.
235. Give the mathematical formula for the variance of a (discrete) random variable X.
236. What is the linearity of expectations?
237. Use the linearity of expectations to calculate the mean of n B (^) p (i.e., solve for the expression of the mean given in class).
238. Use the linearity of expectations to calculate the mean of 7 B. (^) 4.
239. Use the linearity of expectations to calculate the variance of n B (^) p (i.e., solve for the expression of the mean given in class).
240. Use the linearity of expectations to calculate the variance of 7 B. (^) 4.
241. Calculate the mean of 3 B (^) p directly (do not use the linearity of expectations; solve for the expression of the mean given in class).
242. Calculate the mean of 3 B. (^) 3 directly (do not use the linearity of expectations).
243. Calculate the variance of 3 B (^) p directly (do not use the linearity of expectations; solve for the expression of the variance given in class).
244. Calculate the variance of 3 B. (^) 3 directly (do not use the linearity of expectations).
245. Use F, the cumulative probability distribution for a binomial distribution, to express the probability of getting k or fewer successes.
246. Use F, the cumulative probability distribution for a binomial distribution, to express the probability of getting k or more successes.
247. Explain how it is that (^)       k n expresses the number of ways of selecting k objects (unordered) from a set of n objects.
248. What question does (the mass function of the) binomial distribution address? X = 1 3 4 5

probability .3 .4 .2 .1

249. Referring to the table above, calculate the mean of X.
250. Referring to the table above, calculate the variance of X.
251. If X, Y, and Z are i.i.d., and E[X] = 5, E[Y] = 7 and E[Z] = 2, then if W = 4 + 2X +3Y +Z, then what is  for W?
252. If X, Y, and Z are i.i.d., and E[(X – E[X])^2 ] = 2, E[(X – E[X])^2 ] = 1 and E[(X – E[X])^2 ] = 5, then if W = 4 + 2X +3Y +Z, then what is ^ for W?
253. Let F(x) be the cumulative probability function for the distribution 8 B. (^) 4. Calculate F(3).
254. Let F(x) be the cumulative probability function for the distribution 8 B. (^) 4. Calculate the probability of getting a 6 or higher.