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) 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).
[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 ).
(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).
(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.
(4) When the population variances are unknown and know 1 2, then t is referred to t-distribution with df degrees of freedom,
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.
Compute Then, 2 is referred to 2 -distribution with n-1 degrees of freedom.
[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
Compute Then, F is referred to F-distribution with n-1 and m-1 degrees of freedom.
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
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