577. a) From a sample of size $n$, extracted from a normal general population with a known standard deviation $\sigma$, the sample mean $\bar{x}$ has been found. At a significance level $\alpha$, it is required to find the power function of the test criterion for testing the null hypothesis $H_{0}: a=a_{0}$ about the equality of the general mean $a$ to the hypothetical value $a_{0}$ under the alternative hypothesis $H_{1}: a=a_{1} \neq a_{0}$.

**Solution. The alternative hypothesis has the form** $a \neq a_{0}$, therefore the critical region is two-sided. Using rule 1, we find the critical point $u_{\text {cr }}$ from the equality $\Phi\left(u_{\text {cr }}\right)=(1-\alpha) / 2$. Consequently, the two-sided critical region is defined by the inequality $|U|>u_{\mathrm{cr}}$, or in more detail

$$
\left|\frac{\left(\bar{x}-a_{0}\right)}{\sigma / \sqrt{n}}\right|>u_{\mathrm{cr}}
$$

We will find the power of the considered criterion, i.e., the probability of the criterion falling into the critical region assuming that the alternative hypothesis $a=a_{1} \neq a_{0}$ is true:

$$
1-\beta=P\left(\left|\frac{x-a_{0}}{\sigma / \sqrt{n}}\right|>u_{\mathrm{cr}} ; \quad a=a_{1}\right)
$$

Transform the expression under the modulus sign:

$$
\frac{\bar{x}-a_{0}}{\sigma / \sqrt{n}}=\frac{\bar{x}-a_{1}}{\sigma / \sqrt{n}}+\frac{a_{1}-a_{0}}{\sigma / \sqrt{n}}=b+\lambda
$$

where $b=\frac{\bar{x}-a_{1}}{\sigma / \sqrt{n}}, \lambda=\frac{a_{1}-a_{0}}{\sigma / \sqrt{n}}$. Using these relations, we get

$$
\begin{gathered}
1-\beta=P\left(|b+\lambda|>u_{\mathrm{cr}}\right)=P\left(b+\lambda>u_{\mathrm{cr}}\right)+P\left(b+\lambda<-u_{\mathrm{cr}}\right)= \\
=P\left(b>u_{\mathrm{cr}}-\lambda\right)+P\left(b<-u_{\mathrm{cr}}-\lambda\right)= \\
=\left[1-P\left(b<u_{\mathrm{cr}}-\lambda\right)\right]+P\left(b<-u_{\mathrm{cr}}-\lambda\right)= \\
=\left[1-\Phi\left(u_{\mathrm{cr}}-\lambda\right)\right]+\Phi\left(-u_{\mathrm{cr}}-\lambda\right)=1-\Phi\left(u_{\mathrm{cr}}-\lambda\right)-\Phi\left(u_{\mathrm{cr}}+\lambda\right) .
\end{gathered}
$$

Thus, the power of the two-sided criterion at $a=a_{1}$ is

$$
1-\beta=1-\left[\Phi\left(u_{\mathrm{cr}}-\lambda\right)+\Phi\left(u_{\mathrm{cr}}+\lambda\right)\right]
$$

where $\lambda=\left(a_{1}-a_{0}\right) \sqrt{n} / \sigma$.

Each value of $a_{1}$ corresponds to a certain power value, so the power of the criterion is a function of $a_{1}$; denote it by $\pi_{2}\left(a_{1}\right)$.

Thus, the sought power of the two-sided criterion is

$$
\pi_{2}\left(a_{1}\right)=1-\left[\Phi\left(u_{\mathrm{cr}}-\lambda\right)+\Phi\left(u_{\mathrm{cr}}+\lambda\right)\right]
$$

where $\Phi(x)$ is the Laplace function, $\lambda=\left(a_{1}-a_{0}\right) \sqrt{n} / \sigma, u_{\text {cr }}$ is found from the equality $\Phi\left(u_{\text {cr }}\right)=(1-\alpha) / 2$.

b) From a sample of size $n=16$, extracted from a normal general population with a known standard deviation $\sigma=5$, at a significance level of 0.05, the null hypothesis $H_{0}: a=a_{0}=20$ about the equality of the general mean $a$ to the hypothetical value $a_{0}=20$ is tested under the alternative hypothesis $H_{1}: a \neq 20$. Find the power of the two-sided criterion for testing the considered hypothesis for the hypothetical value of the general mean $a_{1}=24$.
Solution. We use the formula

$$
1-\beta=1-\left[\Phi\left(u_{\mathbf{k} p}-\lambda\right)+\Phi\left(u_{\mathbf{k} p}+\lambda\right)\right] .
$$

By rule 1, we find the critical point $u_{\text {cr }}=1.96$.

We calculate $\lambda$, considering that, according to the condition, $a_{1}=24, a_{0}=20$, $n=16, \sigma=5:$

$$
\lambda=\left(a_{1}-a_{0}\right) \sqrt{\bar{n}} / \sigma=(24-20) \sqrt{16} / 5=3.2 .
$$

Substituting $u_{\text {cr }}=1.96$ and $\lambda=3.2$ into formula (*), we get

$$
1-\beta=1-[\Phi(1.96-3.2)+\Phi(1.96+3.2)]=1+\Phi(1.24)-\Phi(5.16) .
$$

From the Laplace function table (see Appendix 2), we find $\Phi(1.24)=0.3925, \Phi(\overline{5}, 16)=0.5$. The desired power $1-\beta=1+$ $+0.3925-0.5=0.8925$.

c) For a sample of size $n=36$, drawn from a normal general population with a known standard deviation $\sigma=6$, the null hypothesis $H_{0}: a=a_{0}=15$ is tested at a significance level of 0.01 against the alternative hypothesis $H_{1}: a \neq a_{0}$. Find the power of the two-sided criterion for testing the considered hypothesis for the hypothetical value of the general mean $a=a_{1}=12$.