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χ 2 Distribution

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?. χ 2 Distribution. χ 2 Distribution.

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χ 2 Distribution

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  1. χ2 Distribution • Given n random independent variants (x1, x2, …, xn), their distribution will be a normal distribution. • If these variants are all squared (x12, x22, …, xn2), will they still fall into a normal distribution?

  2. χ2 Distribution

  3. χ2Distribution • χ2=Σ(x－μ)2/σ2 = ns2/σ2 • df = n－1 • critical value: χα(n)2 • e.g. χ0.05(12)2=21.00

  4. χ2 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: H0: s= σ • χ2=ns2/σ2 =101*15.72/12.12 =170.039 χ2 α/2=129.56 χ 2 >χ2 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, dfb) • degrees of freedom denominator (dfd, dfw)

  8. F distribution

  9. F distribution • F=S2(n1-1)/S2(n2-1) • S2(n1-1):the first variance with the degree of freedom of n1-1 • S2(n2-1): the second variance with the degree of freedom of n2-1

  10. F distribution • Example Sample 1: standard deviation of 19.17, n1=15 Sample 2: standard deviation of 54.19 n2=15 Variance of Sample 1: 19.172=367.49 Variance of Sample 2: 54.192=2936.56 F=367.49/2936.56=7.99 F(14,14, α=0.05)=2.48

  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62，92。方差是否有显著差别？

  12. Case • H0: σ12=σ22 • F=S2(n1-1)/S2(n2-1) = S2(大)/S2(小) ＝ 92 /62 =81/36 =2.25 F(24,30, α=0.05/2)=2.14 F> F(24,30, α=0.05/2) H0 rejected There is significant difference.

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