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

Download Exam B: Statistical Inference and Hypothesis Testing and more Exams Economic statistics in PDF only on Docsity! 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 .950 @qnorm(p) = 1.28 1.31 1.34 1.37 1.41 1.44 1.48 1.51 1.55 1.60 1.64 @tdist(p, 4) = 1.53 1.58 1.62 1.67 1.72 1.78 1.84 1.90 1.97 2.05 2.13 @tdist(p, 5) = 1.48 1.52 1.56 1.60 1.65 1.70 1.75 1.81 1.87 1.94 2.02 @tdist(p, 6) = 1.44 1.48 1.52 1.56 1.60 1.65 1.70 1.75 1.81 1.87 1.94 p = .955 .960 .965 .970 .975 .980 .985 .990 .995 @qnorm(p) = 1.70 1.75 1.81 1.88 1.96 2.05 2.17 2.33 2.58 @tdist(p, 4) = 2.23 2.33 2.46 2.60 2.78 3.00 3.30 3.75 4.60 @tdist(p, 5) = 2.10 2.19 2.30 2.42 2.57 2.76 3.00 3.36 4.03 @tdist(p, 6) = 2.02 2.10 2.20 2.31 2.45 2.61 2.83 3.14 3.71 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 H1 play in the calculations? 9. Suppose a quantile q has been determined for a one-sided hypothesis test where H1: . 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, H0:  = –1, H1:  = 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 H1: . 14. Describe, in ordinary English, what the p-value tells you. 15. In a two-sided hypothesis test, what role, if any, does H1 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, H0:  = –1, H1:  > –1. 18. If you ran a statistical test in Eviews, where H0:  = 60, and H1:  < 60, Eviews would report a larger p-value than is in fact correct. Why? 19. If you ran a statistical test in Eviews, where H0:  = 60, and H1:  > 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 H0:  = 45 and H1:  < 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 H0:  = 45 and H1:  > 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 H0:  = 45 and H1:  ≠ 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 H0? 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. 83. Run and report on a statistical test for the mean where  = .03, H0 :  = 17, H1:  > 17, x = 19.7, s = 4.1, n = 28. 84. Run and report on a statistical test for the mean where  = .03, H0 :  = 17, H1:  ≠ 17, x = 19.7, s = 4.1, n = 28. 85. Run and report on a statistical test for the equality of means where  = .05, H0 : x = y, H1: 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 .950 @qnorm(p) = 1.28 1.31 1.34 1.37 1.41 1.44 1.48 1.51 1.55 1.60 1.64 @tdist(p, 4) = 1.53 1.58 1.62 1.67 1.72 1.78 1.84 1.90 1.97 2.05 2.13 @tdist(p, 12) = 1.36 1.39 1.42 1.46 1.50 1.54 1.58 1.63 1.67 1.73 1.78 p = .955 .960 .965 .970 .975 .980 .985 .990 .995 @qnorm(p) = 1.70 1.75 1.81 1.88 1.96 2.05 2.17 2.33 2.58 @tdist(p, 4) = 2.23 2.33 2.46 2.60 2.78 3.00 3.30 3.75 4.60 @tdist(p, 12) = 1.84 1.91 1.99 2.08 2.18 2.30 2.46 2.68 3.05 Unless specified otherwise, questions about confidence intervals refer to confidence intervals for the mean, with the variance known. Chap 10 1. Give the definition of a consistent estimator. 2. What is the formula for the Law of Large Numbers? 3. In words, what does the Law of Large Numbers say? 4. Explain why the Law of Large Numbers is true 5. Give the definition of a biased estimator. 6. Show that X is an unbiased estimator of . 7. Is     n i i XXn 1 21 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 S2. 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. 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 )(XE 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 {X1,…, X25} is a random sample from N(100, 25), then what is the variance of X ? 54. If {X1,…, X25} 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 1. What kind of thing is a random variable? 2. 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? 3. 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)? 4. 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. 5. 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. 6. 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. 7. 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. 8. Do the same for indexical and time series scales. Chapter 03 9. How would you determine the median of a data set? 10. When is the median an element of a data set? When is it not? 11. Describe how the box of a box-and-whisker plot is constructed. 12. How far out do the whiskers extend? 13. What are the outliers (if any) in a box-and-whisker plot? 14. Know how to determine the three quartile points and the four quartiles of a box-and- whisker plot 15. 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}. 16. What kind of information does a box-and-whisker plot visually display? 17. Give an example of when and how you might use a box-and-whisker plot. 18. How are relative frequency plots different from cumulative frequency plots? 77. For m4, 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 m4, 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 m4? 80. For 1̂ , give: Its name 81. For the standardized skew, give: Its symbol or symbols, if there is more than one 82. For 1̂ , give: Its mathematical formula (using sigma notation) 83. For 1̂ , give: A one sentence characterization of what it measures 84. For 1̂ , give: A brief description of how you could make it larger by altering the data set. 85. For 1̂ , give: A brief description of how you could make it smaller by altering the data set. 86. For 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 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 2̂ , give: Its name 89. For the standardized kurtosis, give: Its symbol or symbols, if there is more than one 90. For 2̂ , give: Its mathematical formula (using sigma notation) 91. For 2̂ , give: A one sentence characterization of what it measures 92. For 2̂ , give: A brief description of how you could make it larger by altering the data set. 93. For 2̂ , give: A brief description of how you could make it smaller by altering the data set. 94. For 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 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, m2) of a data set whose entries have been standardized. 99. What is the relation between 1̂ and m3? 100. What is the relation between 2̂ and m4? 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 m11 = 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 ̂ . 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 E8 pr1 .2 .2 .1 .05 .05 .1 .3 0 pr2 .3 .1 .1 .2 .1 .1 .1 .1 pr3 .3 .05 .04 .05 .1 .1 .1 .1 pr4 .2 .2 .3 .1 –.1 .1 .1 .1 Table 2 Outcomes in the sample space E1 E2 E3 E4 E5 E6 E7 pr .2 .3 .1 .05 .05 .1 .2 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). 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 npB (i.e., solve for the expression of the mean given in class). 238. Use the linearity of expectations to calculate the mean of 74.B . 239. Use the linearity of expectations to calculate the variance of npB (i.e., solve for the expression of the mean given in class). 240. Use the linearity of expectations to calculate the variance of 74.B . 241. Calculate the mean of 3pB directly (do not use the linearity of expectations; solve for the expression of the mean given in class). 242. Calculate the mean of 33.B directly (do not use the linearity of expectations). 243. Calculate the variance of 3pB directly (do not use the linearity of expectations; solve for the expression of the variance given in class). 244. Calculate the variance of 33.B 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 84.B . Calculate F(3). 254. Let F(x) be the cumulative probability function for the distribution 84.B . Calculate the probability of getting a 6 or higher.