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### Lecture3_Part6

Course: ECON 320, Spring 2011
School: SUNY Albany
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Word Count: 745

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Distribution SUNY F at Albany - Department of Economics Eco 320 Economic Statistics F-Distribution 3.0 2.5 F(1, 1) 2.0 1.5 1.0 F(2, 2) 0.5 0.0 0.0 1.0 SUNY at Albany - Department of Economics 2.0 3.0 4.0 5.0 6.0 7.0 Eco 320 Economic Statistics F-Distribution 1.2 1.0 F(2, 10) 0.8 0.6 0.4 F(10, 2) 0.2 0.0 0.0 1.0 SUNY at Albany - Department of Economics 2.0 3.0 4.0 5.0 6.0 7.0...

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Distribution SUNY F at Albany - Department of Economics Eco 320 Economic Statistics F-Distribution 3.0 2.5 F(1, 1) 2.0 1.5 1.0 F(2, 2) 0.5 0.0 0.0 1.0 SUNY at Albany - Department of Economics 2.0 3.0 4.0 5.0 6.0 7.0 Eco 320 Economic Statistics F-Distribution 1.2 1.0 F(2, 10) 0.8 0.6 0.4 F(10, 2) 0.2 0.0 0.0 1.0 SUNY at Albany - Department of Economics 2.0 3.0 4.0 5.0 6.0 7.0 Eco 320 Economic Statistics F-Distribution 1.6 1.4 F(50, 50) 1.2 F(2, 2) 1.0 0.8 0.6 0.4 F(10, 2) 0.2 0.0 0.0 1.0 SUNY at Albany - Department of Economics 2.0 3.0 4.0 5.0 6.0 7.0 Eco 320 Economic Statistics 1. F Distribution A. Let 1 , 2 , , be a random sample of size m from a normal 2 distribution X~N( , ), and let 1 , 2 , , be a random sample of 2 size n from a normal distribution Y~N( , ). Assume that these two 2 2 samples are independent. We want to find out whether = . B. Since we cannot directly observe the two population variances, we use 2 ( )2 2 = ( ) their sample estimator = and 1 1 2 2 Statistical theory shows that if = , the following ratio 2 ( )2 /( 1) = 2 = ( )2 /( 1) 2 C. follows the F distribution with (m-1)(numerator) d.f. and (n-1) (denominator) d.f., designated as 1 ,2 . SUNY at Albany - Department of Economics Eco 320 Economic Statistics 2. Properties of F Distribution A. Properties of the F Distribution F distribution takes only positive values and ranges between 0 and +. It is skewed to the right and approaches the normal distribution as k1 and k2 go to . If W ~ Fm, n, then (1/W) ~ Fn, m. More specifically, 1 F ,m,n F(1 ),n,m 2 = 1, (,) = 2 SUNY at Albany - Department of Economics as n Eco 320 Economic Statistics 3. F Distribution Table F(, k1, k2) SUNY at Albany - Department of Economics Eco 320 Economic Statistics [Example] 1 A. X ~ F(12, 5) with k1 = 12 and k2 = 5. Find 10% critical value of this F distribution. That is, P(X > F(0.10, 12, 5)) = 0.10. SUNY at Albany - Department of Economics Eco 320 Economic Statistics SUNY at Albany - Department of Economics Eco 320 Economic Statistics [Example] 1 A. X ~ F(12, 5) with k1 = 12 and k2 = 5. Find 10% critical of value this F distribution. Since, P(X > 3.27) = 0.10. F(0.10, 12, 5) = 3.27. SUNY at Albany - Department of Economics Eco 320 Economic Statistics [Example] 1 A. X ~ F(12, 5) with k1 = 12 and k2 = 5. Find 10% critical value of this F distribution. Since, P(X > 3.27) = 0.10. F(0.10, 12, 5) = 3.27. P(X > F(0.05, 12, 5)) = 0.05. SUNY at Albany - Department of Economics Eco 320 Economic Statistics SUNY at Albany - Department of Economics Eco 320 Economic Statistics [Example] 1 A. X ~ F(12, 5) with k1 = 12 and k2 = 5. Find 10% critical value of this F distribution. Since, P(X > 3.27) = 0.10. F(0.10, 12, 5) = 3.27 P(X > 4.68) = 0.05. F(0.05, 12, 5) = 4.68 SUNY at Albany - Department of Economics Eco 320 Economic Statistics [Example] 2 A. Illustrate 1 F( ,m,n ) F((1 ), n,m ) F(0.10, 12, 5) = 3.27. F(0.90, 5, 12) = 1/3.27 = 0.306. SUNY at Albany - Department of Economics Eco 320 Economic Statistics [Example] 2 A. Illustrate 1 F( ,m,n ) F((1 ), n,m ) F(0.10, 12, 5) = 3.27. F(0.90, 5, 12) = 1/3.27 = 0.306. 1/ F(0.10, 12, 5) = 1/3.27 = 0.306 = F(0.90, 5, 12). Or, SUNY at Albany - Department of Economics Eco 320 Economic Statistics [Example] 2 A. Illustrate 1 F( ,m,n ) F((1 ), n,m ) F(0.10, 12, 5) = 3.27. F(0.90, 5, 12) = 1/3.27 = 0.306. 1/ F(0.10, 12, 5) = 1/3.27 = 0.306 = F(0.90, 5, 12). Or, 1/ F(0.90, 5, 12) = 1/0.306 = 3.27 = F(0.10, 12, 5). SUNY at Albany - Department of Economics Eco 320 Economic Statistics [Example] A. An instructor gives that same econometrics exam to two classes, one consisting of 100 students and the other consisting of 150 students. She draws a random sample of 25 students from the first class and a random sample of 31 students from the other class and observes that the sample variance of the grade-point average in the two classes are 100 and 132, respectively. It is assumed that the grade-point average in the two classes is normally distributed. Can we assume that the variances of GPA in the two classes are the same? SUNY at Albany - Department of Economics Eco 320 Economic Statistics [In-class Problem] SUNY at Albany - Department of Economics Eco 320 Economic Statistics
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SUNY Albany - ECON - 320
Statistical InferenceSUNY at Albany - Department of EconomicsEco 320 Economic Statistics Inference1. IntroductionA. Population vs. Sample Population: the totality of all possible outcomes of anissue of interest. Sample: a subset of a populationSUN
SUNY Albany - ECON - 320
Regression AnalysisSUNY at Albany - Department of EconomicsEco 320 Economic Statistics Regression1. What is Regression Analysis?SUNY at Albany - Department of EconomicsEco 320 Economic Statistics Regression1. What is Regression Analysis?SUNY at Alb
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Ordinary Least SquaresSUNY at Albany - Department of EconomicsEco 320 Economic Statistics RegressionA. The purpose of regression analysis is to take a purelytheoretical population equation like:Y 0 1 X1 . K X K B. and use a set of sample data to obt
SUNY Albany - ECON - 320
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Hypothesis Testing in Context ofRegression AnalysisSUNY at Albany - Department of EconomicsEco 320 Economic Statistics Regression1. Testing Individual Significance of Regression CoefficientsA. Should Xk be included in the regression equation? Does it
SUNY Albany - ECON - 320
SUNY Albany - ECON - 320
SUNY Albany - ECON - 320
SUMMARYOUTPUTdfRegressionResidualTotalInterceptXVariable1SSMSFSignificanceF1 28808.922 28808.922 29.543531 0.00061948 7801.0776 975.1347936610CoefficientsStandardError tStatPvalue Lower95% Upper95% Lower95.0%Upper95.0%432.41379 16.906074
SUNY Albany - ECON - 320
SUMMARYOUTPUTdfRegressionResidualTotalInterceptXVariable1SSMSFSignificanceF1 28808.922 28808.922 29.543531 0.00061948 7801.0776 975.1347936610CoefficientsStandardError tStatPvalue Lower95% Upper95% Lower95.0%Upper95.0%432.41379 16.906074
SUNY Albany - ECON - 320
TABLE2.6.S&amp;P500IndexandThreeMonthTreasuryBillRate(3MTBill)19802007YearS&amp;P1980118.781981128.051982119.711983160.411984160.461985186.841986236.341987286.831988265.791989322.841990334.591991376.181992415.741993451.411994460.4
SUNY Albany - ECON - 320
TABLE2.6.S&amp;P500IndexandThreeMonthTreasuryBillRate(3MTBill)19802007YearS&amp;P1980118.781981128.051982119.711983160.411984160.461985186.841986236.341987286.831988265.791989322.841990334.591991376.181992415.741993451.411994460.4
SUNY Albany - ECON - 320
Table27MedianHomePriceandMortgageInterestRateintheU.S.,19802007Year1980198019801980198119811981198119821982198219821983198319831983198419841984198419851985198519851986198619861986198719871987198719881988198819881989
SUNY Albany - ECON - 320
Table27MedianHomePriceandMortgageInterestRateintheU.S.,19802007Year1980198019801980198119811981198119821982198219821983198319831983198419841984198419851985198519851986198619861986198719871987198719881988198819881989
SUNY Albany - ECON - 320
Table28HypotheticalDataonWeeklyConsumptionExpenditureandWeeklyIncomeWeeklyIncome WeeklyConsumptionExpenditure(\$)(X)80100120140160180200220240260(\$)(Y)55,60,65,70,7565,70,74,80,85,8879,84,90,94,9880,93,95,103,108,113,115102,107,110,116,
SUNY Albany - ECON - 320
Table28HypotheticalDataonWeeklyConsumptionExpenditureandWeeklyIncomeWeeklyIncome WeeklyConsumptionExpenditure(\$)(X)80100120140160180200220240260(\$)(Y)55,60,65,70,7565,70,74,80,85,8879,84,90,94,9880,93,95,103,108,113,115102,107,110,116,
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SUNY Albany - ECON - 320
SUNY Albany - ECON - 370
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SUNY Albany - ECON - 370
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ASU - INT - 111
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ASU - INT - 111
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