Chapter 8.3, Problem 61E

a.

To determine

To find: The critical value.

Answer to Problem 61E

The critical value is 2.262

Explanation of Solution

Given:

Confidence level = 95%

Sample size ( n ) = 10

Calculation:

Level of significance $\left(\alpha \right)=1-0.95=0.05$

The degree of freedom can be calculated as:

  $\begin{array}{c}df=n-1\\ =10-1\\ =9\end{array}$

Using t -critical value table, the critical value at 5% level of significance can be calculated as:

  $\begin{array}{c}{t}_{\alpha /2,df}=z_{0.05/2,9}\\ =2.262\end{array}$

Thus, the critical value is 2.62.

b.

To determine

To find: The critical value.

Answer to Problem 61E

The critical value is 2.861.

Explanation of Solution

Given:

Confidence level = 99%

Sample size ( n ) = 20

Calculation:

Level of significance $\left(\alpha \right)=1-0.99=0.01$

Degree of freedom can be calculated as:

  $\begin{array}{c}df=n-1\\ =20-1\\ =19\end{array}$

The critical value at 1% level of significance can be calculated as:

  $\begin{array}{c}{t}_{\alpha /2,df}=t_{0.01/2,19}\\ =2.861\end{array}$

Thus, the critical value is 2.861.

c.

To determine

To find: The critical value.

Answer to Problem 61E

The critical value is 1.671.

Explanation of Solution

Given:

Confidence level = 90%

Sample size ( n ) = 77

Calculation:

Level of significance $\left(\alpha \right)=1-0.90=0.10$

Degree of freedom can be calculated as:

  $\begin{array}{c}df=n-1\\ =77-1\\ =76\end{array}$

The critical value at 10% level of significance can be calculated as:

  $\begin{array}{c}{t}_{\alpha /2,df}=t_{0.10/2,76}\\ =1.671\end{array}$

Thus, the critical value is 1.671.