Question

Critical values 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. $A99\%$ confidence interval from an SRS of $20$ observations

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

Short answer

a. The critical value is $2.262$

b. The critical value is $2.861$

c. The critical value is$1.671$

Step-by-step solution

Part (a) Step 1: Given Information

It is given that Confidence Level $=95\%$

$n=10$

Part (a) Step 2: Calculation

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

Degree of $df=n-1$

$=10-1=9$

At $5\%$, level of significance

$t_{\alpha /2,df}=z_{0.05/2,9}=2.262$

The critical value is$2.262$

Part (b) Step 1: Given Information

It is given that Confidence Level $=99\%$

$n=20$

Part (b) Step 2: Calculation

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

Degree of freedom, $df=n-1$

$=20-1=19$

At $1\%$level of significance,

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

The critical value is$2.861$

Part (c) Step 5: Given Infromation

It is given that Confidence Level $=90\%$

$n=77$

Part (c) Step 2: Calculation

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

Degree of freedom, $df=n-1$

$=77-1=76$

At $10\%$level of significance, critical value is

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

The critical value is$1.671$