Bernhard Kabelka - Mathematical Statistics: "Testing - Special tests for the normal distribution"

Special tests for the normal distribution

known
- $H_{0}:\,{µ <= µ_{0}\;/\;H_1:\,{µ > µ_{0}$ : reject if $\overlineX_n> µ_{0}+ z_{1-alpha} * \sqrt{sigma^2/n}$
- $H_{0}:\,{µ >= µ_{0}\;/\;H_1:\,{µ < µ_{0}$ : reject if $\overlineX_n> µ_{0}- z_{1-alpha} * \sqrt{sigma^2/n}$
- $H_{0}:\,{µ = µ_{0}\;/\;H_1:\,{µ <> µ_{0}$ : reject if $|\overlineX_n- µ_{0}| > z_{1-\frac{alpha}2 * \sqrt{sigma^2/n}$
unknown
- $H_{0}:\,{µ <= µ_{0}\;/\;H_1:\,{µ > µ_{0}$ : reject if $\overlineX_n> µ_{0}+ t_{n-1;1-alpha} * \sqrt{S_n^2/n}$
- $H_{0}:\,{µ >= µ_{0}\;/\;H_1:\,{µ < µ_{0}$ : reject if $\overlineX_n> µ_{0}- t_{n-1;1-alpha} * \sqrt{S_n^2/n}$
- $H_{0}:\,{µ = µ_{0}\;/\;H_1:\,{µ <> µ_{0}$ : reject if $|\overlineX_n- µ_{0}| > t_{n-1;1-\frac{alpha}2 * \sqrt{S_n^2/n}$

known
- $H_{0}:\,{sigma^2<= sigma^2_{0}\;/\;H_1:\,{sigma^2> sigma^2_{0}$ : reject if $sum (X_i - µ)^2 > sigma^2_{0}* chi_{n;1-alpha}$
- $H_{0}:\,{sigma^2>= sigma^2_{0}\;/\;H_1:\,{sigma^2< sigma^2_{0}$ : reject if $sum (X_i - µ)^2 < sigma^2_{0}* chi_{n;alpha}$
- $H_{0}:\,{sigma^2= sigma^2_{0}\;/\;H_1:\,{sigma^2<> sigma^2_{0}$ : reject if $sum (X_i - µ)^2 < sigma^2_{0}* chi_{n;\frac{alpha}2$ or if $sum (X_i - µ)^2 > sigma^2_{0}* chi_{n;1-\frac{alpha}2$
unknown
- $H_{0}:\,{sigma^2<= sigma^2_{0}\;/\;H_1:\,{sigma^2> sigma^2_{0}$ : reject if $S_n^2> \frac{sigma^2_{0}{n-1} * chi_{n-1;1-alpha}$
- $H_{0}:\,{sigma^2>= sigma^2_{0}\;/\;H_1:\,{sigma^2< sigma^2_{0}$ : reject if $S_n^2< \frac{sigma^2_{0}{n-1} * chi_{n-1;alpha}$
- $H_{0}:\,{sigma^2= sigma^2_{0}\;/\;H_1:\,{sigma^2<> sigma^2_{0}$ : reject if $S_n^2< \frac{sigma^2_{0}{n-1} * chi_{n-1;\frac{alpha}2$ or if $S_n^2> \frac{sigma^2_{0}{n-1} * chi_{n-1;1-\frac{alpha}2$

Tests for equality of means:

If and are known, reject if
$|\overlineX_n- \overline{Y......{sigma^2_1n + \frac{sigma^2_2m$
and if the variance are not known, but are at least equal, reject if
$|\overlineX_n- \overline{Y......-1)S_X^2+(m-1)S_Y^2)}$
where and denote the sample variances of the samples and respectively.
Test for equality of the variances:

Reject if
$\frac{S_X^2{S_Y^2 > F_{......^2{S_Y^2 < F_{n-1,m-1;\frac{alpha}2$