1. Hypothesis testing for the mean [A] One population that follows a normal distribution H 0 :  =  0 vs H 1 :    0 Suppose that we collect independent data, x 1, x 2, …, x n ~ N( ,  2 ).

2. (2) When the population variance is unknown, use t-test i.e., replace the population variance with the sample variance and then t is referred to the t-distribution with n-1 degrees of freedom. (1) When the population variance is known, use z-test then z is referred to N(0,1).

3. [B] Two-normal-population case H 0 :  1 =  2 vs H 1 :  1   2 Assume that we collect independent data, x 11, x 21, …, x n1 ~ N(  1,  1 2 ) and x 12, x 22, …, x m2 ~ N(  2,  2 2 ).

4. (1) When the population variances are known and  1 =  2, then z is referred to N(0,1). (2) When the population variances are known and  1   2 then z is referred to N(0,1).

5. (3) When the population variances are unknown but know  1 =  2, where then t is referred to t-distribution with n+m-2 degrees of freedom. Note: s 2 is called pooled sample variance.

6. (4) When the population variances are unknown and know  1   2, then t is referred to t-distribution with df degrees of freedom,

7. Hypothesis testing for the variance [A] Assume that we collect independent data, x 1, x 2, …, x n ~ N( ,  2 ). Want to test H 0 :  2 =  0 2 vs H 1 :  2   0 2.

8. Compute Then,  2 is referred to  2 -distribution with n-1 degrees of freedom.

9. [B] Assume that we collect independent data, x 11, x 21, …, x n1 ~ N(  1,  1 2 ) and x 12, x 22, …, x m2 ~ N(  2,  2 2 ). Want to test H 0 :  1 2 =  2 2 vs H 1 :  1 2   2 2

10. Compute Then, F is referred to F-distribution with n-1 and m-1 degrees of freedom.

11. Analysis of variance (ANOVA) [A] One-way ANOVA Assume that we collect independent data, x 11, x 21, …, x n1 ~ N(  1,  2 ), x 12, x 22, …, x m2 ~ N(  2,  2 ), …, x 1k, x 2k, …, x pk ~ N(  k,  2 ). Want to test H 0 :  1 =  2 = …=  k vs H 1 : not H 0

12. We may rephrase the problem x i j =    j +  i j,  i j ~ N(0,  2 ), the hypotheses can be rewritten as H 0 :  1 =  2 = …=  k = 0 vs H 1 : not H 0 One-way ANOVA is a statistical model to test the above H 0 vs H 1