Chapter 8.2, Problem 3CYU

a.

To determine

Calculate the critical value and the critical region at the significance level of $\alpha =0.05$.

Answer to Problem 3CYU

The critical value at $\alpha =0.05$ is $\underset{\_}{\pm 1.96}$ and the critical region at $\alpha =0.05$ is $\underset{\_}{z\le -1.96\text{or}z\ge 1.96}$.

Explanation of Solution

It is given that the value of test statistic z as 1.78 and significance level $\alpha =0.05$.

Denote $\mu$ as the true population mean.

The given test hypotheses are:

Null hypothesis: $H_0:\mu =12$.

Alternate hypothesis: $H_1:\mu \ne 12$.

From the given hypotheses, the alternative hypothesis contains the not equal $(\ne )$ symbol. Thus, it is clear that the hypotheses follow the two-tailed test.

From Table 8.1 “Table of Critical Values”, the critical value for two-tailed test at $\alpha =0.05$ level is $\underset{\_}{\pm 1.96}$.

Thus, the critical region is $\underset{\_}{z\le -1.96\text{or}z\ge 1.96}$.

b.

To determine

Decide whether the null hypothesis $H_0$ is rejected at $\alpha =0.05$ level.

Answer to Problem 3CYU

The null hypothesis $H_0$ is not rejected at $\alpha =0.05$ level.

Explanation of Solution

It is given that the test statistic $z=1.78$ and the significance level $\alpha =0.05$.

From previous part (a), the critical value is $\pm 1.96$.

Decision based on the critical value method:

For left-tailed test: If $z\le -z_{\alpha }$, reject $H_0$.

For right-tailed test: If $z\ge z_{\alpha }$, reject $H_0$.

For two-tailed test: If $z\le -z_{\frac{\alpha }{2}}\text{or}z\ge z_{\frac{\alpha }{2}}$, reject $H_0$.

Conclusion:

The critical values at $\alpha =0.05$ level are $\pm 1.96$ and the test statistic is 1.78.

Here, the test statistic value of 1.78lies within the critical values of $\pm 1.96$.

That is, $-z_{\frac{\alpha }{2}}\left(-1.96\right)<z\left(=1.78\right)<z_{\frac{\alpha }{2}}\left(1.96\right)$.

Therefore, the null hypothesis is not rejected.

c.

To determine

Obtain the critical value and the critical region for a significance level of $\alpha =0.01$.

Answer to Problem 3CYU

The critical value at $\alpha =0.01$ is $\underset{\_}{\pm 2.576}$ and the critical region at $\alpha =0.01$ is $\underset{\_}{z\ge 2.576}$

Explanation of Solution

It is given that the significance level $\alpha =0.01$.

From Table 8.1 “Table of Critical Values”, the critical value for two-tailed test at $\alpha =0.01$ level is $\underset{\_}{\pm 2.576}$.

Thus, the critical region is $\underset{\_}{z\le -2.576\text{or}z\ge 2.576}$.

d.

To determine

Decide whether the null hypothesis $H_0$ is rejected at $\alpha =0.01$ level.

Answer to Problem 3CYU

The null hypothesis $H_0$ is not rejected at $\alpha =0.01$ level.

Explanation of Solution

It is given that the test statistic $z=1.78$ and the significance level $\alpha =0.01$.

From previous part (c), the critical value is $\pm 2.576$.

Decision based on the critical value method:

For left-tailed test: If $z\le -z_{\alpha }$, reject $H_0$.

For right-tailed test: If $z\ge z_{\alpha }$, reject $H_0$.

For two-tailed test: If $z\le -z_{\frac{\alpha }{2}}\text{or}z\ge z_{\frac{\alpha }{2}}$, reject $H_0$.

Conclusion:

The critical value at $\alpha =0.01$ level is $\pm 2.576$ and the test statistic is 1.78.

Here, the test statistic value of 1.78 is lies within the critical values of $\pm 2.576$.

That is, $-z_{\frac{\alpha }{2}}\left(-2.576\right)<z\left(=1.78\right)<z_{\frac{\alpha }{2}}\left(2.576\right)$.

Therefore, the null hypothesis is not rejected.