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Midterm Review

Midterm Review. Econ 240A. The Big Picture. The Classical Statistical Trail. Rates & Proportions. Inferential Statistics. Application. Descriptive Statistics. Discrete Random Variables. Binomial. Probability. Power 4-#4. Discrete Probability Distributions; Moments.

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Midterm Review

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  1. Midterm Review Econ 240A

  2. The Big Picture

  3. The Classical Statistical Trail Rates & Proportions Inferential Statistics Application Descriptive Statistics Discrete Random Variables Binomial Probability Power 4-#4 Discrete Probability Distributions; Moments

  4. Descriptive StatisticsPower One-Lab One • Concepts central tendency: mode, median, mean dispersion: range, inter-quartile range, standard deviation (variance) • Are central tendency and dispersion enough descriptors?

  5. Draw a Histogram Concepts • Normal Distribution • Central tendency: mean or average • Dispersion: standard deviation • Non-normal distributions

  6. The Classical Statistical Trail Rates & Proportions Descriptive Statistics Inferential Statistics Application Classicall Modern Discrete Random Variables Binomial Probability Power 4-#4 Discrete Probability Distributions; Moments

  7. Exploratory Data Analysis • Stem and Leaf Diagrams • Box and Whiskers Plots

  8. Weight Data Males: 140 145 160 190 155 165 150 190 195 138 160 155 153 145 170 175 175 170 180 135 170 157 130 185 190 155 170 155 215 150 145 155 155 150 155 150 180 160 135 160 130 155 150 148 155 150 140 180 190 145 150 164 140 142 136 123 155 Females: 140 120 130 138 121 125 116 145 150 112 125 130 120 130 131 120 118 125 135 125 118 122 115 102 115 150 110 116 108 95 125 133 110 150 108

  9. Box Diagram median First or lowest quartile; 25% of observations below Upper or highest quartile 25% of observations above

  10. 3rd Quartile + 1.5* IQR = 156 + 46.5 = 202.5; 1st value below =195

  11. The Classical Statistical Trail Rates & Proportions Inferential Statistics Application Descriptive Statistics Discrete Random Variables Binomial Probability Power 4-#4 Discrete Probability Distributions; Moments

  12. Power Three - Lab Two • Probability

  13. Operations on events • The event A and the event B both occur: • Either the event A or the event B occurs or both do: • The event A does not occur, i.e.not A:

  14. Probability statements • Probability of either event A or event B • if the events are mutually exclusive, then • probability of event B

  15. Conditional Probability • Example: in rolling two dice, what is the probability of getting a red one given that you rolled a white one? • P(R1/W1) ?

  16. In rolling two dice, what is the probability of getting a red one given that you rolled a white one?

  17. Conditional Probability • Example: in rolling two dice, what is the probability of getting a red one given that you rolled a white one? • P(R1/W1) ?

  18. Independence of two events • p(A/B) = p(A) • i.e. if event A is not conditional on event B • then

  19. The Classical Statistical Trail Rates & Proportions Inferential Statistics Application Descriptive Statistics Discrete Random Variables Binomial Probability Power 4-#4 Discrete Probability Distributions; Moments

  20. Power 4 – Lab Two

  21. Three flips of a coin; 8 elementary outcomes 3 heads 2 heads 2 heads 1 head 2 heads 1 head 1 head 0 heads

  22. The Probability of Getting k Heads • The probability of getting k heads (along a given branch) in n trials is: pk *(1-p)n-k • The number of branches with k heads in n trials is given by Cn(k) • So the probability of k heads in n trials is Prob(k) = Cn(k) pk *(1-p)n-k • This is the discrete binomial distribution where k can only take on discrete values of 0, 1, …k

  23. Expected Value of a discrete random variable • E(x) = • the expected value of a discrete random variable is the weighted average of the observations where the weight is the frequency of that observation

  24. Variance of a discrete random variable • VAR(xi) = • the variance of a discrete random variable is the weighted sum of each observation minus its expected value, squared,where the weight is the frequency of that observation

  25. Lab Two • The Binomial Distribution, Numbers & Plots • Coin flips: one, two, …ten • Die Throws: one, ten ,twenty • The Normal Approximation to the Binomial • As n ∞, p(k) N[np, np(1-p)] • Sample fraction of successes:

  26. Lab Three and Power 5,6 Z~N(0,1) Prob(-1.96≤z≤1.96)=0.95 2.5% 2.5% -1.96 1.96

  27. Hypothesis Testing: Rates & Proportions One-tailed test: Step #2: test statistic One-tailed test: Step #1:hypotheses One-tailed test: Step #3: choose  e.g.  = 5% Z=1.645 Reject if Step # 4: this determines The rejection region for H0 5%

  28. Remaining Topics • Interval estimation and hypothesis testing for population means, using sample means • Decision theory • Regression • Estimators • OLS • Maximum lilelihood • Method of moments • ANOVA

  29. Midterm Review Cont. Econ 240A

  30. Last Time

  31. The Classical Statistical Trail Rates & Proportions Inferential Statistics Application Descriptive Statistics Discrete Random Variables Binomial Probability Power 4-#4 Discrete Probability Distributions; Moments

  32. Remaining Topics • Interval estimation and hypothesis testing for population means, using sample means • Decision theory • Regression • Estimators • OLS • Maximum lilelihood • Method of moments • ANOVA

  33. Lab Three Power 7 Population Random variable x Distribution f(m, s2) f ? Pop. Sample Sample Statistic Sample Statistic:

  34. f(x) in this example is Uniform X~U(0.5, 1/12) E(x) = 0.5 Var(x) = 1/12 Nonetheless, from the central Limit theorem, the sample mean Has a normal density f(x) 0 x 1

  35. Histogram of 50 Sample Means, Uniform, U(0.5, 1/12) Average of the 50 sample means: 0.4963

  36. Inference Z=-1.96 Z=1.96 2.5% 2.5%

  37. Confidence Intervals • If the population variance is known, use the normal distribution and z • If the population variance is unknown, use Student’s t-distribution and t

  38. t-distribution Text p.253 Normal compared to t t distribution as smple size grows

  39. Appendix B Table 4 p. B-9

  40. Hypothesis tests Step two: choose the test statistic Step One: state the hypotheses You choose v 2-tailed test Step four: reject the null hypothesis if the test statistic is in the Rejection region Step Three: choose the size Of the Type I error, =0.05 Z=-1.96 Z=1.96 2.5% 2.5%

  41. True State of Nature p = 0.5 P > 0.5 No Error 1 - a Type II error b C(II) Accept null Decision Type I error a C(I) No Error 1 - b Reject null E[C] = C(I)* a + C(II)* b

  42. Regression Estimators • Minimize the sum of squared residuals • Maximum likelhood of the sample • Method of moments

  43. Minimize the sum of squared residuals

  44. Maximum likelihood Method of moments

  45. Inference in RegressionInterval estimation

  46. Estimated Coefficients, Power 8

  47. Appendix B Table 4 p. B-9

  48. Inference in RegressionHypothesis testing Step One State the hypothesis Step Two Choose the test statistic Step Four Reject the null hypothesis if the Test statistic is in the rejection region Step Three Choose the size of the Type I error, 

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