1. 1 Midterm Review Econ 240A

2. 2 The Big Picture

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

4. 4 Descriptive Statistics Power 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. 5 Concepts Normal Distribution –Central tendency: mean or average –Dispersion: standard deviation Non-normal distributions Draw a Histogram

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

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

8. 8 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 Weight Data

9. 9

10. 10 Box Diagram First or lowest quartile; 25% of observations below Upper or highest quartile 25% of observations above median

11. 11 3 rd Quartile + 1.5* IQR = 156 + 46.5 = 202.5; 1 st value below =195

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

13. 13 Power Three - Lab Two Probability

14. 14 Operations on events The event A and the event B both occur: Either the event A or the event B occurs or both do: Either the event A or the event B occurs or both do: The event A does not occur, i.e.not A:

15. 15 Probability statements Probability of either event A or event B –if the events are mutually exclusive, then probability of event B

16. 16 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) ?

17. 17 In rolling two dice, what is the probability of getting a red one given that you rolled a white one?

18. 18 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) ?

19. 19 Independence of two events p(A/B) = p(A) –i.e. if event A is not conditional on event B –then

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

21. 21 Power 4 – Lab Two

22. 22 Three flips of a coin; 8 elementary outcomes 3 heads 2 heads 1 head 2 heads 1 head 0 heads

23. 23 The Probability of Getting k Heads The probability of getting k heads (along a given branch) in n trials is: p k *(1-p) n-k The number of branches with k heads in n trials is given by C n (k) So the probability of k heads in n trials is Prob(k) = C n (k) p k *(1-p) n-k This is the discrete binomial distribution where k can only take on discrete values of 0, 1, …k

24. 24 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 the expected value of a discrete random variable is the weighted average of the observations where the weight is the frequency of that observation

25. 25 Variance of a discrete random variable VAR(x i ) = 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 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

26. 26 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:

27. 27 1.96-1.96 Z~N(0,1) Prob(- 1.96≤z≤1.96)=0.95 2.5% Lab Three and Power 5,6

28. 28 Hypothesis Testing: Rates & Proportions One-tailed test: Step #1:hypotheses One-tailed test: Step #2: test statistic One-tailed test: Step #3: choose  e.g.  = 5% Z=1.645 5% Step # 4: this determines The rejection region for H 0 Reject if

29. 29 Remaining Topics Interval estimation and hypothesis testing for population means, using sample means Decision theory Regression –Estimators OLS Maximum lilelihood Method of moments –ANOVA

30. 30 Midterm Review Cont. Econ 240A

31. 31 Last Time

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

33. 33 Remaining Topics Interval estimation and hypothesis testing for population means, using sample means Decision theory Regression –Estimators OLS Maximum lilelihood Method of moments –ANOVA

34. 34 Population Random variable x Distribution f(    f ? Sample Sample Statistic: Sample Statistic Pop. Lab Three Power 7

35. 35 x 0 1 f(x) 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

36. 36 Histogram of 50 Sample Means, Uniform, U(0.5, 1/12) Average of the 50 sample means: 0.4963

37. 37 Inference Z=1.96 2.5% Z=-1.96 2.5%

38. 38 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

39. 39 Text p.253 Normal compared to t t-distribution t distribution as smple size grows

40. 40 Appendix B Table 4 p. B-9

41. 41 Hypothesis tests Step One: state the hypotheses Step two: choose the test statistic Step Three: choose the size Of the Type I error, =0.05 Z=1.96 2.5% Z=-1.96 2.5% Step four: reject the null hypothesis if the test statistic is in the Rejection region 2-tailed test You choose v

42. 42 Decision Accept null Reject null True State of Nature p = 0.5P > 0.5 No Error 1 -  Type I error  C(I) No Error 1 -  Type II error  C(II) E[C] = C(I)*  + C(II)* 

43. 43 Regression Estimators Minimize the sum of squared residuals Maximum likelhood of the sample Method of moments

44. 44 Minimize the sum of squared residuals

45. 45 Maximum likelihood Method of moments

46. 46 Inference in Regression Interval estimation

47. 47 Estimated Coefficients, Power 8 Coefficients Standard Errort StatP-valueLower 95% Upper 95% Intercept-1.023777760.727626534-1.407010.167999648-2.4994727620.451917 X Variable 10.065650260.00108632860.433168.58311E-380.0634470850.067853

48. 48 Appendix B Table 4 p. B-9

49. 49 Inference in Regression Hypothesis testing Step One State the hypothesis Step Two Choose the test statistic Step Three Choose the size of the Type I error,  Step Four Reject the null hypothesis if the Test statistic is in the rejection region