Hypothesis testing for the mean A One population

(1) When the population variance is known, use z-test then z is referred to

[B] Two-normal-population case H 0 : 1 = 2 vs H 1: 1 2

(1) When the population variances are known and 1 = 2 , then z

(3) When the population variances are unknown but know 1 = 2 , where

(4) When the population variances are unknown and know 1 2 , then t

Hypothesis testing for the variance [A] Assume that we collect independent data, x 1,

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, …, xn 1

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

We may rephrase the problem xi j = j + i j, i j

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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, …, xn ~ N( , 2).

(1) When the population variance is known, use z-test then z is referred to N(0, 1). (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.

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

(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, …, xn ~ N( , 2). Want to test H 0 : 2 = 02 vs H 1: 2 02.

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, …, xn 1 ~ N( 1, 12) and x 12, x 22, …, xm 2 ~ N( 2, 22). Want to test H 0 : 12 = 22 vs H 1: 12 22

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, …, xn 1 ~ N( 1, 2), x 12, x 22, …, xm 2 ~ N( 2, 2), …, x 1 k, x 2 k, …, xpk ~ N( k, 2). Want to test H 0 : 1 = 2 = …= k vs H 1: not H 0

We may rephrase the problem xi 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