Question

𝛼 = 0.01; two-tailed test; 𝑛 = 15; 𝜎 = 5 .

Answer 1

The following null and alternative hypotheses need to be tested:

$H_0:\mu=\mu_0$

$H_0:\mu\not=\mu_0$

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is $\alpha=0.01,$ and the critical value for a two-tailed test is $z_c=2.5758.$

The rejection region for this two-tailed test is $R=\{z:|z|>2.5758\}.$

The z-statistic is computed as follows:

$$z=\dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$$

If it is observed that $|z|>z_c,$ it is then concluded that the null hypothesis is rejected. There is enough evidence to claim that the population $\mu$ is different than $\mu_0,$ at the $\alpha=0.01$ significance level.

If it is observed that $|z|\leq z_c,$ it is then concluded that the null hypothesis is not rejected. There is not enough evidence to claim that the population $\mu$  is different than $\mu_0,$ at the $\alpha=0.01$ significance level.