1. χ 2 Distribution Given n random independent variants (x 1, x 2, …, x n ), their distribution will be a normal distribution. If these variants are all squared (x 1 2, x 2 2, …, x n 2 ), will they still fall into a normal distribution?

2. χ 2 Distribution

3. χ 2 = Σ (x － μ ) 2 / σ 2 = ns 2 / σ 2 df = n － 1 critical value: χ α(n) 2 e.g. χ 0.05(12) 2 =21.00

4. χ 2 Distribution Common usage: Inference on a single normal variance. Chi-squared Tests: - test for independence, - homogenity, - goodness of fit.

5. Case 教材试用：中等水平的学校 人数： n=101 标准差： s=15.7 全区标准差： σ=12.1 试点平均分＝全区平均分 α= 0.05 教材是否适用？

6. Case Null hypothesis: H 0 : s= σ χ 2 =ns 2 / σ 2 =101*15.7 2 /12.1 2 =170.039 χ 2 α/2 =129.56 χ 2 > χ 2 α/2 Conclusion: suitable for the advanced students, but not for the intermediate.

7. F distribution The F distribution is the distribution of the ratio of two estimates of variance. It is used to compute probability values in the analysis of variance. The F distribution has two parameters: degrees of freedom numerator (dfn, df b ) degrees of freedom denominator (dfd, df w )

8. F distribution

9. F=S 2 (n1-1) /S 2 (n2-1) S 2 (n 1 -1) : the first variance with the degree of freedom of n 1 -1 S 2 (n2-1) : the second variance with the degree of freedom of n 2 -1

10. Example Sample 1: standard deviation of 19.17, n 1 =15 Sample 2: standard deviation of 54.19 n 2 =15 Variance of Sample 1: 19.17 2 =367.49 Variance of Sample 2: 54.19 2 =2936.56 F=367.49/2936.56=7.99 F( 14,14, α=0.05 )=2.48 F distribution

11. Case 两个班使用不同的教学方法，甲班 31 人， 乙班 25 人。期末两个班考试成绩方差分 别为 6 2 ， 9 2 。方差是否有显著差别？

12. Case H 0 : σ 1 2 =σ 2 2 F=S 2 (n1-1) /S 2 (n2-1) = S 2 ( 大 ) /S 2 ( 小 ) ＝ 9 2 / 6 2 =81/36 =2.25 F (24,30, α=0.05/2) =2.14 F> F (24,30, α=0.05/2) H 0 rejected There is significant difference.