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Hypothesis testing for the mean

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 ). (1) When the population variance is known, use z-test

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Hypothesis testing for the mean

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  1. Hypothesis testing for the mean [A] One population that follows a normal distribution H0 :  = 0 vs H1:   0 Suppose that we collect independent data, x1, x2, …, xn ~ N( , 2).

  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.

  3. [B] Two-normal-population case H0 : 1 = 2 vs H1: 1  2 Assume that we collect independent data, x11, x21, …, xn1 ~ N(1, 12) and x12, x22, …, xm2 ~ N(2, 22).

  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: s2 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, x1, x2, …, xn ~ N( , 2). Want to test H0 : 2 = 02 vs H1: 2  02.

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

  9. [B] Assume that we collect independent data, x11, x21, …, xn1 ~ N(1, 12) and x12, x22, …, xm2 ~ N(2, 22). Want to test H0 : 12 = 22 vs H1: 12  22

  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, x11, x21, …, xn1 ~ N(1, 2), x12, x22, …, xm2 ~ N(2, 2), …, x1k, x2k, …, xpk ~ N(k, 2). Want to test H0 : 1 = 2 = …= k vs H1: not H0

  12. We may rephrase the problem xi j =   j + i j, i j ~ N(0, 2), the hypotheses can be rewritten as H0 : 1 =  2 = …=  k = 0 vs H1: not H0 One-way ANOVA is a statistical model to test the above H0 vs H1

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