Question

What critical value $t*$from Table B should be used for a confidence interval for the population mean in each of the following situations?

a. A $95\%$confidence interval based on $n=10$randomly selected observations

b. A $99\%$confidence interval from an SRS of $20$observations

c. A $90\%$confidence interval based on a random sample of $77$ individuals

Short answer

Part a. The critical value is $2.262$.

Part b. The critical value is $2.861$

Part c. The critical value is$1.671$

Step-by-step solution

Part a. Step 1. Given information

Confidence level $=95\%$

Sample size$\left(n\right)=10$

Part a. Step 2. Explanation

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

The degree of freedom can be calculated as:

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

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

$$\begin{gathered} t_{\alpha /2,df}=z_{0.05/2.9} \\ =2.262 \end{gathered}$$

Thus, the critical value is$2.262$

Part b. Step 1. Given information

Confidence level = $99\%$

Sample size $\left(n\right)=20$

Part b. Step 2. Explanation

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

Degree of freedom can be calculated as:

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

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

$$\begin{gathered} t_{\alpha /2,df}=t_{0.01/2,19} \\ =2.861 \end{gathered}$$

Thus, the critical value is$2.861$

Part c. Step 1. Given information

Confidence level = $90\%$

Sample size $\left(n\right)=77$

Part c. Step 2. Explanation

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

Degree of freedom can be calculated as:

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

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

$$\begin{gathered} t_{\alpha /2,df}=t_{0.10,76} \\ =1.671 \end{gathered}$$

Thus, the critical value is$1.671$.