Chapter 8.5, Problem 1E

Using Table G, find the critical value(s) for each. Indicate the noncritical region or regions, and state the null and alternative hypotheses. Use σ2 = 225.

a. α = 0.10, n = 14, two-tailed

b. α = 0.05, n = 27, right-tailed

c. α = 0.01, n = 9, left-tailed

d. α = 0.05, n = 17, right-tailed

To determine

To find: The critical values for two tailed test.

Answer to Problem 1E

The critical values for two tailed test is 5.892 and 23.362, respectively.

Null hypothesis:

$H_0:{\sigma }^2=225$

Alternative hypothesis:

$H_1:{\sigma }^2\ne 225$

Explanation of Solution

Given info:

The level of significance is $\alpha =0.10$, the sample size is 14 and the test is two-tailed.

Answer:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =14-1\\ =13\end{array}$

The area to the right of larger value is, $\frac{0.10}{2}=0.05$ and the area to right of the smaller value is $1-\frac{0.10}{2}=0.95$.

From “Table G: The chi square distribution”, the critical values for the $\alpha$ value in the table of 0.05 and 0.95 with 14 degrees of freedom is 23.362 and 5.892, respectively.

Critical and non critical region:

Software Procedure:

Step-by-step procedure to obtain the critical region and non critical using the MINITAB software:

• Choose Graph > Probability Distribution Plot > choose View Probability> OK.
• From Distribution, choose Chi-square.
• In Degrees of freedom, enter 13.
• Click the Shaded Area tab.
• Choose Probability value and Both Tail for the region of the curve to shade.
• Enter the Probability value as 0.10.
• Click OK.

Output using the MINITAB software is given below:

State the null and alternative hypotheses:

Here, the claim is that the population variance is 225. This can be written as ${\sigma }^2=225$. The complement of the claim is, ${\sigma }^2\ne 225$. In the given experiment, the null hypothesis indicates the claim.

Null hypothesis: $H_0:{\sigma }^2=225$

Alternative hypothesis: $H_1:{\sigma }^2\ne 225$.

To determine

To find: The critical value for right tailed test.

Answer to Problem 1E

The critical value for right tailed test is 38.885.

Null hypothesis:

$H_0:{\sigma }^2=225$

Alternative hypothesis:

$H_1:{\sigma }^2>225$.

Explanation of Solution

Given info:

The level of significance is $\alpha =0.05$, the sample size is 27 and the test is right-tailed.

Answer:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =27-1\\ =26\end{array}$

From “Table G: The chi square distribution”, the critical values for the $\alpha$ value in the table of 0.05 with 26 degrees of freedom is 38.885.

Critical and non critical region:

Software Procedure:

Step-by-step procedure to obtain the critical region and non critical using the MINITAB software:

• Choose Graph > Probability Distribution Plot > choose View Probability> OK.
• From Distribution, choose Chi-square.
• In Degrees of freedom, enter 26.
• Click the Shaded Area tab.
• Choose Probability value and Right Tail for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Click OK.

Output using the MINITAB software is given below:

State the null and alternative hypotheses:

Null hypothesis: $H_0:{\sigma }^2=225$

Alternative hypothesis: $H_1:{\sigma }^2>225$

To determine

To find: The critical value for left tailed test.

Answer to Problem 1E

The critical value for left tailed test is 1.646.

Null hypothesis:

$H_0:{\sigma }^2=225$

Alternative hypothesis:

$H_1:{\sigma }^2<225$

Explanation of Solution

Given info:

The level of significance is $\alpha =0.01$, the sample size is 9 and the test is left-tailed.

Answer:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =9-1\\ =8\end{array}$

If the test is left tail, the level of significance is subtracted from 1. That is, $1-0.01=0.99$.

From “Table G: The chi square distribution”, the critical values for the $\alpha$ value in the table of 0.99 with 8 degrees of freedom is 1.646.

Critical and non critical region:

Software Procedure:

Step-by-step procedure to obtain the critical region and non critical using the MINITAB software:

• Choose Graph > Probability Distribution Plot > choose View Probability> OK.
• From Distribution, choose Chi-square.
• In Degrees of freedom, enter 8.
• Click the Shaded Area tab.
• Choose Probability value and Left Tail for the region of the curve to shade.
• Enter the Probability value as 0.01.
• Click OK.

Output using the MINITAB software is given below:

State the null and alternative hypotheses:

Null hypothesis: $H_0:{\sigma }^2=225$

Alternative hypothesis: $H_1:{\sigma }^2<225$

To determine

To find: The critical value for right tailed test.

Answer to Problem 1E

The critical value for right tailed test is 26.296.

Null hypothesis:

$H_0:{\sigma }^2=225$

Alternative hypothesis:

$H_1:{\sigma }^2>225$

Explanation of Solution

Given info:

The level of significance is $\alpha =0.05$, the sample size is 17 and the test is right-tailed.

Answer:

Degrees of freedom:

$\begin{array}{c}df=n-1\\ =17-1\\ =16\end{array}$

From “Table G: The chi square distribution”, the critical values for the $\alpha$ value in the table of 0.05 with 16 degrees of freedom is 26.296.

Critical and non critical region:

Software Procedure:

Step-by-step procedure to obtain the critical region and non critical using the MINITAB software:

• Choose Graph > Probability Distribution Plot > choose View Probability> OK.
• From Distribution, choose Chi-square.
• In Degrees of freedom, enter 16.
• Click the Shaded Area tab.
• Choose Probability value and Right Tail for the region of the curve to shade.
• Enter the Probability value as 0.05.
• Click OK.

Output using the MINITAB software is given below:

State the null and alternative hypotheses:

Null hypothesis: $H_0:{\sigma }^2=225$

Alternative hypothesis: $H_1:{\sigma }^2>225$