Korean, Edit

Chapter 14-7. Chi-squared Test (3 types)

Higher category : 【Statistics】 Chapter 14. Statistcal Test 


1. Chi-square goodness-of-fit test

2. Chi-square test of independence

3. Chi-square test of homogeneity

4. Chi-Square distribution and central limit theorem



drawing

Figure 1. chi-squared distribution table


1. chi-square goodness-of-fit test

⑴ problem situation (contingency table): if the situation is genetic experiment, Oi and Ej are not proportions but numbers of population


drawing

Figure 2. problem situation of chi-square goodness-of-fit test


⑵ setting hypotheses

H0 : the sample Xi follows the given probability distribution

H1 : the sample Xi does not follow the given probability distribution

⑶ calculation of test statistic


drawing


⑷ critical region: significance level is α. it is well established when the sample size is big 


drawing


⑸ Chi-square goodness-of-fit test for multiple variables is also possible.


  1 2 S Total
X(1) N1(1) N2(1) NS(1) n(1)
X(2) N1(2) N2(2) NS(2) n(2)
X(K) N1(K) N2(K) NS(K) n(K)
  m1 m2 mS n

Table 1. Chi-square goodness-of-fit test for multiple variables


스크린샷 2025-03-13 오후 11 49 27



2. chi-square test of independence 

⑴ problem situation (contingency table)


drawing

Figure 3. problem situation of chi-square test of independence


⑵ setting hypotheses 

H0 : X and Y are independent

H1 : X and Y are not independent

⑶ calculation of test statistic: calculate the predicted value by using marginal probability distribution and the relationship of independence 


drawing


⑷ critical region: significance level is α. it is well established when the sample size is big 


drawing



3. chi-square test of homogeneity 

⑴ problem situation (contingency table) 


drawing

Figure 4. problem situation of chi-square test of homogeneity


⑵ setting hypotheses 

H0 : Group1 and Group2 follow the same probability distribution

H1 : Group1 and Group2 don’t follow the same probability distribution

⑶ calculation of test statistic

① equal probability distribution is clearly different from independence conceptually 

② however, chi-square test of homogeneity follows a very similar calculation to a chi-square test of independence


drawing


⑷ critical region: significance level is α. it is well established when the sample size is big 


drawing



4. Chi-Square distribution and central limit theorem

⑴ Central limit theorem (CLT)


스크린샷 2025-03-03 오전 8 42 52


⑵ Derivation of the Chi-Square distribution: Using the fact that for Xi following a binomial distribution, the variance is Var(Xi) = npq = np - np2 ≈ np = 𝔼[Xi], we get…


스크린샷 2025-03-03 오전 8 43 51



Input : 2019.10.05 11:13

results matching ""

    No results matching ""