Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch.19 A Significance Test 1.Principles of Statistical Hypothesis Test 2.Null Hypothesis and Alternative Hypothesis 3.Test Statistic and Significance Level 4.Type-1 Error and Type-2 Error 5.Significance Test Process number box: Repetition in a Large Number number box: Repetition in a Small Number 8.t - test Appendix : Using the Internet (Baseball Statistic)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/26 INDEX 1 Principles of Statistical Hypothesis Test 2 Null Hypothesis and Alternative Hypothesis 3 Test Statistic and Significance Level 4 Type-1 Error and Type-2 Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/26 1.Principles of Statistical Hypothesis Test Significance Test Ex) Tax Revenue under the new taxation ‘Is there no fluctuation on Tax revenue’ Population: 0.1mil. Of Tax payment data Sample: 100 of Random Sample from the Population Difference in Tax payment under new Taxation compared to the old one Sample Average :KRW(-) 0.219mil. Sample S.D.:KRW0.725mil. ? Does This Difference occur by chance or by real factor? Does This Difference occur by chance or by real factor?

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/26 Principles of Significance Test-Reduction to absurdity 1.Principles of Statistical Hypothesis Test  100times Random drawing out from a box of 0.1mil.  Each Number on cards mean the difference of tax At first, Suppose that Average of box is “ 0 ”  S.E. of Sample Average  -KRW0.219mil.-KRW0 73,000 = -3 Probability of Occurring (- )KRW0.219 difference by chance is only 1/1,000

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/26 INDEX 1 Principles of Statistical Hypothesis Test 2 Null Hypothesis and Alternative Hypothesis 3 Test Statistic and Significance Level 4 Type-1 Error and Type-2 Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/26  Null Hypothesis ( H 1 ) Difference occurred by chance Average of Box is “ o ”  Alternative Hypothesis( H 2 ) Difference is result of real factor Average of Box is a negative 2. Null Hypothesis and Alternative Hypothesis Null and Alternative Hypothesis Express Hypothesis by ‘ notation on Box Model ’

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/26 INDEX 1 Principles of Statistical Hypothesis Test 2 Null Hypothesis and Alternative Hypothesis 3 Test Statistic and Significance Level 4 Type-1 Error and Type-2 Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/26 Observed – Expect S.E. of Observed Value z or t = 3. Test statistic and Significance Level Test Statistic  Measuring the difference between Observed Value of Data and Expect Value under Null  Popular z -statistic or t -statistic Under the Null Using Sample Standard Deviation

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/26 Observed Significance Level : p -value 3. Test statistic and Significance Level mil ,000 z -statistic in former Ex. = = -3 Provability less then -3 in S.E. unit = 1/1000 This probability is called ‘ Observed Significance Level ’ or ‘ p-value ’ p -value means probability earning a extreme test statistic above observed statistic. As This probability get less, Base of argument against Null get larger.

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/26 Reject of Null Hypothesis 3. Test statistic and Significance Level  Z -test is demonstration by a contradiction Null( H 0 ) p -value is less than certain  Reject the Null! Significance Level (Base of decision)  p -value <  : Rejecting H 0 in a  significance level (Statistical significant)  p -value >  : Not to Reject H 0 in a  significance level (Statistical insignificant)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/26 One-sided Test and Two-sided Test 3. Test statistic and Significance Level Alternative( H 1 ) Null( H 0 ) Argument of a Alternative has a direction ☞ One-sided Test Not having a direction ☞ Two-sided Test Ex) One-sided Two-sided Vs

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/26 INDEX 1 Principles of Statistical Hypothesis Test 2 Null Hypothesis and Alternative Hypothesis 3 Test Statistic and Significance Level 4 Type-1 Error and Type-2 Error

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/26 4. Type-1 Error and Type-2 Error Type-1 Error and Type-2 Error Reality Test Result Null is TrueAlternative is true Not Rejecting a NullProper DecisionType-2 Error Rejecting a NullType-1 ErrorProper Decision Power of Test = 1- Prob.(Type-2 Error) Probability to make proper decision in a case that we should reject the Null ( ‘ This makes you reject when you should ’ =power)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/26 INDEX 5 Significance Test Process number box: Repetition in a Large Number number box: Repetition in a Small Number 8 t-test Appendix : Using the Internet (Baseball Statistics)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/26 5. Significance Test Process Significance Test Process Design a Box Model and Make a Null hypothesis Calculate a Expect Value given that Null hypothesis is true Define a Test Statistic measuring difference between a Observed and a Expect Compare p -value to significance level  Calculate a p -value, the observed significance level

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 16/26 INDEX 5 Significance Test Process number box: Repetition in a Large Number number box: Repetition in a Small Number 8 t-test Appendix : Using the Internet (Baseball Statistics)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 17/ number box: Repetition in a Large Number Example ‘ Karl ’ is known as a Great Predictor on fluctuation of Stock Price Index  Making him predict directions of Stock index fluctuation during 100 commercial days  He hit 65days during 100 days Is Hitting Extra 15 days over 50days by his ability or just by chance?

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 18/ number box: Repetition in a Large Number Ex. of Z -test Z 검정 H 0 : Karl has a special ability 0 1 It is like drawing cards 100 times in replacement randomly 0 and 1, Each has prob.50% Observed – Expect S.E. of Observed S.D. of Box = S.E. of Number = It can not be just by chance p -value==0.14% 3

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 19/26 INDEX 5 Significance Test Process number box: Repetition in a Large Number number box: Repetition in a Small Number 8 t-test Appendix : Using the Internet (Baseball Statistics)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 20/26 When we have small sample (1) : Z -test number box: Repetition in a Small Number Ex) Date proposal of ‘ Karl ’  1 date per 2 proposal (before)  1 date per 9 proposal (now) ‘ Did Ann change her mind? ’ : Ann still loves Karl 0 1 Sum of 9times random replacement sampling from box above  Because of small sample, We can not apply Normal Distribution value=1%

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 21/26 When we have small sample (2) : Sign Test When We have small sample, Qualitative data from 0-1 box should be applied to Sign Test number box: Repetition in a Small Number  Under the Null hypothesis, Number of having date (X) follows the Binomial Distribution

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 22/26 INDEX 5 Significance Test Process number box: Repetition in a Large Number number box: Repetition in a Small Number 8 t-test Appendix : Using the Internet (Baseball Statistics)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 23/26 Example 8. t - test Ex) Accuracy of Speedometer  Select 5 speedometers in random  Measure a car ’ s speed running at 100 km/h 108, 113, 98, 102, 118  Average of Observed data =  Standard Deviation of Observed data = 7.22  S.D of Sample Sum =  S.E of Sample Aver. =  H 0 : There is no bias in each speedometer

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 24/26 t - test 8. t - test  p -value when using Normal Distribution Curve < 1% Because We have a small sample, we can ’ t calculate an accurate Standard Deviation of the box So, Normal Distribution Curve can ’ t be applied  Use t -distribution curve Because We have a small sample, we can ’ t calculate an accurate Standard Deviation of the box So, Normal Distribution Curve can ’ t be applied  Use t -distribution curve value Degree of Freedom = ‘ Number of Observation-1 ’ = 5-1= 4 As D.F. get less, Tail get thicker

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 25/26 INDEX 5 Significance Test Process number box: Repetition in a Large Number number box: Repetition in a Small Number 8 t-test Appendix : Using the Internet (Baseball Statistics)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 26/26 Appendix : Using the Internet (Baseball Statistics)  In general, People think a ‘ Homerun Hitter ’ as a Big Man with powerful swing. So People think a ‘ Homerun Hitter ’ has many strike-knock-outs and slow feet.  Do a Empirical Work on that myth. Baseball Statistics – Myth Vs Statistics - Baseball Statistics – Myth Vs Statistics -