Chapter 8, Problem 20CAP

1

To determine

Identify the decision for each obtained value if $\alpha =0.05$

1. a) $z_{\text{obt}}=2.10$
2. b) $z_{\text{obt}}=1.70$
3. c) $z_{\text{obt}}=2.75$
4. d) $z_{\text{obt}}=-3.30$

Answer to Problem 20CAP

1. a) The decision for $z_{\text{obt}}=2.10$ if $\alpha =0.05$ is rejecting the null hypothesis.
2. b) The decision for $z_{\text{obt}}=1.70$ if $\alpha =0.05$ is rejecting the null hypothesis.
3. c) The decision for $z_{\text{obt}}=2.75$ if $\alpha =0.05$ is rejecting the null hypothesis.
4. d) The decision for $z_{\text{obt}}=-3.30$ if $\alpha =0.05$ is retaining the null hypothesis.

Explanation of Solution

Decision Rule:

If the value of the test statistic is greater than the critical value then reject the null hypothesis.

a)

The data gives the value of test statistic value is 2.10, the test is upper-tail test and $\alpha =0.05$.

From the “Table 8.4”, the critical value corresponding with 0.05 level and upper-tailed test is 1.645.

Justification: The test statistic value is 2.10 and the critical value is 1.645. That is, the test statistic is greater than the critical value. From the decision rule, the null hypothesis is rejected.

Hence, the decision for $z_{\text{obt}}=2.10$ if $\alpha =0.05$ is rejecting the null hypothesis.

b)

The test statistic value is 1.70 and the critical value is 1.645.

Justification: The test statistic value is greater than the critical value. That is, $z_{\text{obt}}\left(=1.70\right)>\text{critical value}\left(=1.645\right)$. From the decision rule, the null hypothesis is rejected.

Hence, the decision for $z_{\text{obt}}=1.70$ if $\alpha =0.05$ is rejecting the null hypothesis.

c)

The test statistic value is 2.75 and the critical value is 1.645.

Justification: The test statistic value is greater than the critical value. That is, $z_{\text{obt}}\left(=2.75\right)>\text{critical value}\left(=1.645\right)$. From the decision rule, the null hypothesis is rejected.

Hence, the decision for $z_{\text{obt}}=2.75$ if $\alpha =0.05$ is rejecting the null hypothesis.

d)

The test statistic value is –3.30 and the critical value is 1.645.

Justification: The test statistic value is lesser than the critical value. That is, $z_{\text{obt}}\left(=-3.30\right)\text{<critical value}\left(=1.645\right)$. From the decision rule, the null hypothesis is not rejected.

Hence, the decision for $z_{\text{obt}}=-3.30$ if $\alpha =0.05$ is retaining the null hypothesis.

2

To determine

Identify the decision for each of the obtained value if $\alpha =0.01$

1. a) $z_{\text{obt}}=2.10$
2. b) $z_{\text{obt}}=1.70$
3. c) $z_{\text{obt}}=2.75$
4. d) $z_{\text{obt}}=-3.30$

Answer to Problem 20CAP

1. a) The decision for $z_{\text{obt}}=2.10$ if $\alpha =0.01$ is rejecting the null hypothesis.
2. b) The decision for $z_{\text{obt}}=1.70$ if $\alpha =0.01$ is rejecting the null hypothesis.
3. c) The decision for $z_{\text{obt}}=2.75$ if $\alpha =0.01$ is rejecting the null hypothesis.
4. d) The decision for $z_{\text{obt}}=-3.30$ if $\alpha =0.01$ is rejecting the null hypothesis.

Explanation of Solution

Decision Rule:

If the test statistic is greater than the upper critical value or the test statistic is lesser than the lower critical value, then reject the null hypothesis.

a)

The data gives the value of test statistic as 2.10, the test is two-tail test and $\alpha =0.01$.

From the “Table 8.4”, the critical value corresponding with 0.01 level and two-tailed test is $\pm 2.58$.

Justification: The test statistic value is 2.10 and the critical values are –2.58 and 2.58. That is, the test statistic (=2.10) is less than the upper critical value (=2.58) and the test statistic value (=2.10) is greater than lower critical value (=–2.58). From the decision rule, the null hypothesis is not rejected.

Hence, the decision for $z_{\text{obt}}=2.10$ if $\alpha =0.01$ is retaining the null hypothesis.

b)

The data gives the value of test statistic value is 1.70 and the critical values are –2.58 and 2.58.

Justification: The test statistic (=1.70) is less than the upper critical value (=2.58) and the test statistic value (=1.70) is greater than lower critical value (=–2.58). From the decision rule, the null hypothesis is not rejected.

Hence, the decision for $z_{\text{obt}}=1.70$ if $\alpha =0.01$ is retaining the null hypothesis.

c)

The test statistic value is 2.75 and the critical values are –2.58 and 2.58.

Justification: The test statistic value (=2.75) is greater than the upper critical value (=2.58). That is, $z_{\text{obt}}\left(=2.75\right)>\text{critical value}\left(=2.58\right)$. From the decision rule, the null hypothesis is rejected.

Hence, the decision for $z_{\text{obt}}=2.75$ if $\alpha =0.01$ is rejecting the null hypothesis.

d)

The test statistic value is –3.30 and the critical values are –2.58 and 2.58.

Justification: The test statistic value (=–3.30) is lesser than the lower critical value (=–2.58). That is, $z_{\text{obt}}\left(=-3.30\right)\text{<critical value}\left(=-2.58\right)$. From the decision rule, the null hypothesis is rejected.

Hence, the decision for $z_{\text{obt}}=-3.30$ if $\alpha =0.01$ is rejecting the null hypothesis.