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Ch. 22  2 - test

Ch. 22  2 - test. Introduction to  2 - test Structure of  2 – test Testing Stochastic Independence. 3. 1. 2. Introduction to 2 - test. Structure of 2 – test. INDEX. Testing Stochastic Independence. 1. Introduction to 2 - test. Usage of - test.

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Ch. 22  2 - test

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  1. Ch. 22 2 - test Introduction to 2 - test Structure of 2– test Testing Stochastic Independence

  2. 3 1 2 Introduction to 2 - test Structure of 2– test INDEX Testing Stochastic Independence

  3. 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

  4. 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

  5. 1. Introduction to 2 - test Usage of - 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 Drawing out Cards having numbers from 1 to 6 on each other from a box with replacement z – test t - test • Testing the Null : aver. of box is 3.5 • Testing the Null : the prob. one card drawn out is 1/6 each - test

  6. 1. Introduction to 2 - test An Ex. of - test • Does a Gambler use a unfair die? Result from 60 times casting 4 3 3 1 2 3 4 6 5 6 2 4 1 3 3 5 3 4 3 4 3 3 4 5 4 5 6 4 5 1 6 4 4 2 3 3 2 4 4 5 6 3 6 2 4 6 4 6 3 2 5 4 6 3 3 3 5 3 1 4 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

  7. = (observed-expect)2 expect 1. Introduction to 2 - test - 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. The bigger -statistic means there is big difference between Observed values and Expect values.

  8. 1. Introduction to 2 - test Usage of - test • 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

  9. 1. Introduction to 2 - test Degree of freedom of - test 2–distribution curve responding to D.F.(5) and D.F.(10) 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 • That distribution curves are right-tailed. • As D.F. get larger, Shape of curve get more symmetric as moving to right.  D.F. = 6-1 = 5

  10. 1. Introduction to 2 - test - distribution curve 2-distribution curve in D.F.(5) Read the probability area in the first column of table. p-value = -statistics table : a section 14.2 면적과 자유도가 만나는 위치에 놓인 수치를 읽는다. 11.07 5% critical value 15.09 1% critical value The size of area right to 14.2 is the value between 5% and 1%

  11. 3 1 2 Introduction to 2 - test Structure of 2– test INDEX Testing on Stochastic Independence

  12. 2. Structure of 2– test Structure Basic Data Stochastic Model A Frequency Table 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 Recording frequencies of each observation And making the result as a kind of table 1

  13. (observed-expect)2 expect 2. Structure of 2– test Structure Observed Significance level (p-value) 2-statistics 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 Ex) 6-1=5 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% 0

  14. 3 1 2 Introduction to 2 - test Structure of 2– test INDEX Testing Stochastic Independence

  15. 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 3.Testing Stochastic Independence Test for Stochastic Independence among variables • Is it stochastic independent? : Left-handedness and Gender? Gender and a Preferred hand (ratio) Gender and a Preferred hand (frequency)

  16. ? ? ? ? ? ? Right-handed Male Right-handed Female Left-handed Male Left-handed Female Ambidexter Male Ambidexter Female 3.Testing Stochastic Independence Designing a box model • 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

  17. 3.Testing Stochastic Independence Null vs Alternative Observed and Expect per each category (Calculation of Expect will be following) Calculate Expect values under the Null.

  18. -statistic 3.Testing Stochastic Independence 2 - test 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). • Degree of Freedom 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

  19. 3.Testing Stochastic Independence 2 - test • p-value 2-distribution curve of D.F.(2) 0.2% p-value 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.

  20. 3.Testing Stochastic Independence Expected Frequencies (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.

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