Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch. 22  2 - test 1. Introduction to  2 - test 2. Structure of  2 – test 3. Testing Stochastic Independence

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 2/20 INDEX 3 Testing Stochastic Independence 1 Introduction to  2 - test 2 Structure of  2 – test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 3/20 1. Introduction to  2 - test Usage of - test  Predicting whether Stock price index would be up or down: There are only 2 categories z – test Sign test  Predicting level of Stock price index by intervals: There are categories more than 2  2 – test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 4/20  If Average of cards in Box being only matter … z – test t - test  If the number of several kinds of cards in box being matter … – test 1. Introduction to  2 - test Usage of - test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 5/20 - test  Testing the Null : aver. of box is 3.5 z – test t - test  Testing the Null : the prob. one card drawn out is 1/6 each Drawing out Cards having numbers from 1 to 6 on each other from a box with replacement   2 -test indicates whether we can consider observed sample as from random sampling when we know about composition of contents in box  z -test or t -test indicate whether we can consider observed sample as from random sampling when we only know average of box 1. Introduction to  2 - test Usage of - test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 6/20  Does a Gambler use a unfair die? Result from 60 times casting NumberObservedExpect 합 60 Result from 60 times drawing out cards having numbers from 1 to 6 on each with replacement from a box The Observed is much larger than the Expect. 1. Introduction to  2 - test An Ex. of - test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 7/20 - statistic  Only one or two ridiculous columns can not determine whether whole data ’ s ridiculousness.  There needs certain indicators presenting overall difference between the observed and the expect getting all information together. = (observed-expect) 2 expect The bigger -statistic means there is big difference between Observed values and Expect values. 1. Introduction to  2 - test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 8/20  The earned value, 14.2 is too big to think the model is true.  It may be possible to earn such a large number when casting a fair die in 60 times, but the size of possibility matters.  Earn 1,000 of  2- statistics by 1,000 times repetition of casting a fair die 60 times and then calculating the  2- statistic.  When applying  2- statistics to a histogram (in fact, a Empirical Histogram of  2-distribution), the Area of histogram right to the value  The ratio of 1,000 개의  2 -statistics to 1,000 statistics more than 14.2 The  2- statistics more than 14.2 are strong evidences against the model.  How big the probability would be that One stochastic model produce such a strong contrary evidence against itself ?  Meaning of p-value The  2- statistics more than 14.2 are strong evidences against the model.  How big the probability would be that One stochastic model produce such a strong contrary evidence against itself ?  Meaning of p-value 1. Introduction to  2 - test Usage of - test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 9/20 Degree of freedom of - test  2 – distribution curve responding to D.F.(5) and D.F.(10)  That distribution curves are right-tailed.  As D.F. get larger, Shape of curve get more symmetric as moving to right.  That distribution curves are right-tailed.  As D.F. get larger, Shape of curve get more symmetric as moving to right. As Model is designed in the concrete,It is meaningless to infer the population parameter :D.F. = the number of terms used in calculating  2 -statistic - 1  D.F. = 6-1 = 5 1. Introduction to  2 - test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 10/20 - distribution curve -statistics table : a section 자유도 50%10%5%1%  The size of area right to 14.2 is the value between 5% and 1%  2 -distribution curve in D.F.(5) Read the probability area in the first column of table p -value = 면적과 자유도가 만 나는 위치에 놓인 수 치를 읽는다 % critical value % critical value 1. Introduction to  2 - test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 11/20 INDEX 3 Testing on Stochastic Independence 1 Introduction to  2 - test 2 Structure of  2 – test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 12/20 2. Structure of  2 – test Structure In general, Size of sample is represented as n Ex) n =60 Box Model Ex.) a Die Model: A box containing Cards having numbers 1~6 on each Random Sampling with replacement from a composition Announced box Basic DataStochastic Model Recording frequencies of each observation And making the result as a kind of table A Frequency Table 1 1

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 13/20 Structure In the case of no need to infer the population parameter, D.F. is as below the number of terms used in calculating  2-statistic - 1 Ex) 6-1=5  2 -statistics Degree of Freedom The p-value is the size of area right to  2- statistic under the  2 -distribution curve of corresponding D.F. Ex) p-value=1.4% Observed Significance level ( p -value) (observed-expect) 2 expect Structure of  2 – test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 14/20 INDEX 3 Testing Stochastic Independence 1 Introduction to  2 - test 2 Structure of  2 – test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 15/20 3.Testing Stochastic Independence Test for Stochastic Independence among variables  Is it stochastic independent? : Left-handedness and Gender? MaleFemale Right9341,070 Left11392 Ambidexter208 M(100%)F(100%) Right87.5%91.4% Left10.6%7.9% Ambidexter1.9%0.7% Gender and a Preferred hand (frequency) Gender and a Preferred hand (ratio) [Physiology] As Women ’ s left brain is more activated than Men ’ s, More Right-handedness. [Sociology] Women got forced more to use Right hand than men. The Ratio of preferred hand is Identical to both Men and Women, Difference above is just by chance It is by ChanceIt is by Real

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 16/20 3.Testing Stochastic Independence Designing a box model ? Right-handed Male ? Right-handed Female ? Left-handed Male ? Left-handed Female ? Ambidexter Male ? Ambidexter Female  Make a Box model under the assumption that 2,237 people of sample are randomly drawn out from population. 2,237 times of Random Sampling without replacement MaleFemale Right9341,070 Left11392 Ambidexter208

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 17/20 3.Testing Stochastic Independence Null vs Alternative Observed FrequencyExpected Frequency MaleFemaleMaleFemale Right9341, Left Ambidexter Gender and a Preferred hand Difference in ratio between Gender and a Preferred hand NullMutually Independent Just a coincidence occurred during sampling process AlternativeA practical relation exists Reflects practical difference of population Calculate Expect values under the Null. Observed and Expect per each category (Calculation of Expect will be following)

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 18/20 3.Testing Stochastic Independence  2 - test  -statistic  Degree of Freedom MaleFemaleSum Right Left Ambidexter7-70 Sum000 Difference between Observed and Expect per each category As two values are given, the rests will be determined automatically : Only two deviations are free among 6 D.F. = (3-1)  (2-1) = 2 When testing stochastic independence on a m  n table, If there is no probability restriction except stochastic independence, the D.F. will be ( m -1)  ( n -1).

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 19/20 3.Testing Stochastic Independence  p -value  2 -distribution curve of D.F.(2) 12  0.2% p -value  자유도 2 인 In  2 -distribution curve of D.F.(2), Size of the area right to 12 is 0.2%. So. Reject the Null.  We can tell Gender and a preferred hand : mutually dependent.  2 - test

Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics STATISTICS 20/20 3.Testing Stochastic Independence Expected Frequencies Observed Frequencies Ratio Expected Frequencies MaleFemaleMaleFemale Right9341, %9561,048 Left %98107 Ambidexter2081.3%1315 Sum1,0671,170100%1,0671,170 (934+1,070)/2,237  89.6% : If gender and a preferred hand were mutually independent, Number of right-handed male is expected to be 956 (89.6% of the 1,067 male)  Getting the Expect using both Sample data and Null hypothesis.  As Getting the expect by inference, this results in reduction of D.F.