Transcript statistic
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 INDEX STATISTICS 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 2/20 STATISTICS 1. Introduction to 2 - test Usage of - test 2 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 3/20 STATISTICS 1. Introduction to 2 - test Usage of - test 2 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… 2– test Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 4/20 STATISTICS 1. Introduction to 2 - test Usage of - test 2 Drawing out Cards having numbers from 1 to 6 on each other from a box with replacement z – test Testing the Null : aver. of box is 3.5 t - test Testing the Null : the prob. one card drawn out is 1/6 each 2 - test 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 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 5/20 STATISTICS 1. Introduction to 2 - test 2 An Ex. of - test Does a Gambler use a unfair die? Result from 60 times casting Number Observed Expect 4331234656 1 4 10 2413353434 2 6 10 3345456451 3 17 10 4 16 10 5 8 10 6 9 10 합 60 60 6442332445 6362464632 5463335314 The Observed is much larger than the Expect. Result from 60 times drawing out cards having numbers from 1 to 6 on each with replacement from a box Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 6/20 1. Introduction to 2 - test STATISTICS 2- 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. 2 = 2-statistic means The bigger there is big difference between Observed values and Expect values. (observed-expect)2 expect (4 10) 2 (6 10) 2 (17 10) 2 (16 10) 2 (8 10) 2 (9 10) 2 14.2 10 10 10 10 10 10 2 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 7/20 STATISTICS 1. Introduction to 2 - test Usage of - test 2 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 14.2. The ratio of 1,000개의 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 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 8/20 1. Introduction to 2 - test STATISTICS Degree of freedom of 2- test 2 –distribution curve responding to D.F.(5) and D.F.(10) 20 That distribution curves are right-tailed. 자유도5 자유도10 % As D.F. get larger, Shape of curve get more symmetric as moving to right. 10 0 0 5 10 15 20 25 30 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 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 9/20 STATISTICS 1. Introduction to 2 - test -2 distribution curve 2-distribution curve in D.F.(5) Read the probability area in the first column of table. p-value = 14.2 면적과 자유도가 만 나는 위치에 놓인 수 11.07 15.09 치를 읽는다. 5% critical 1% critical value value The size of area right to 14.2 is the value between 5% and 1% 2-statistics table : a section 자유도 50% 10% 5% 1% 3 2.37 6.25 7.82 11.34 4 3.36 7.78 9.49 13.28 5 4.35 9.24 11.07 15.09 6 5.35 10.65 12.59 16.80 7 6.35 12.02 14.07 18.48 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 10/20 INDEX STATISTICS 1 Introduction to 2 - test 2 Structure of 2 – test 3 Testing on Stochastic Independence Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 11/20 STATISTICS 2. Structure of 2 – test Structure Basic Data Stochastic Model 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 A Frequency Table Recording frequencies of each observation And making the result as a kind of table Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 12/20 2. Structure of 2 – test STATISTICS Structure 2-statistics (observed-expect)2 expect Degree of Freedom 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 Observed Significance level (p-value) 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% Ex) 6-1=5 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 13/20 INDEX STATISTICS 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 14/20 3.Testing Stochastic Independence STATISTICS Test for Stochastic Independence among variables Is it stochastic independent? : Left-handedness and Gender? Gender and a Preferred hand (frequency) Male Female Right 934 1,070 Left 113 Ambidexter 20 Gender and a Preferred hand (ratio) M(100%) F(100%) Right 87.5% 91.4% 92 Left 10.6% 7.9% 8 Ambidexter 1.9% 0.7% It is by Real It is by Chance [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 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 15/20 3.Testing Stochastic Independence STATISTICS Designing a box model Make a Box model under the assumption that 2,237 people of sample are randomly drawn out from population. ? Right-handed Male ? Right-handed Female ? Left-handed Male ? Left-handed Female ? Ambidexter Male ? Ambidexter Female Male Female Right 934 1,070 Left 113 92 Ambidexter 20 8 2,237 times of Random Sampling without replacement Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 16/20 3.Testing Stochastic Independence STATISTICS Null vs Alternative Gender and a Preferred hand Difference in ratio between Gender and a Preferred hand Null Mutually Independent Just a coincidence occurred during sampling process Alternative A practical relation exists Reflects practical difference of population Observed and Expect per each category (Calculation of Expect will be following) Observed Frequency Calculate Expect values under the Null. Expected Frequency Male Female Male Female Right 934 1,070 956 1.048 Left 113 92 98 107 Ambidexter 20 8 13 15 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 17/20 3.Testing Stochastic Independence STATISTICS 2 - test 2 -statistic (934 956) 2 (1,070 1,048) 2 (113 98) 2 (92 107) 2 (20 13) 2 (8 15) 2 12 956 1,048 98 107 13 15 2 Degree of Freedom Difference between Observed and Expect per each category As two values are given, the rests will be determined automatically : Male Female Sum Right -22 22 0 Left 15 -15 0 Ambidexter 7 -7 0 Only two deviations are free among 6 Sum 0 0 0 D.F. = (3-1)(2-1) = 2 When testing stochastic independence on a mn 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 18/20 3.Testing Stochastic Independence STATISTICS 2-2 test p-value 2-distribution curve of D.F.(2) p-value 0.2% 12 자유도 2인 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. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 19/20 3.Testing Stochastic Independence STATISTICS Expected Frequencies Observed Frequencies Male Female Right 934 1,070 Left 113 Ambidexter Sum Ratio Expected Frequencies Male Female 89.6% 956 1,048 92 9.1% 98 107 20 8 1.3% 13 15 1,067 1,170 100% 1,067 1,170 (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. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 20/20