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 $90\%$ confidence interval based on $n=12$ randomly selected observations

b. A $95\%$ confidence interval from an SRS of $30$ observations

c. A $99\%$ confidence interval based on a random sample of size $58$

Short answer

a. The critical value is $1.796$.

b. The critical value is $2.045$.

c. The critical value is$2.678$.

Step-by-step solution

Part (a) Step 1: Given Information

It is given that $n=12$

Confidence Level$=0.90$

Part (a) Step 2: Calculation

Degree of freedom is calculated as:

$df=n-1=12-1=11$

Level of Significance, $a=0.10$

EXCEL formula $=TINV(0.1,11)$gives

${\mathrm{t}}_{11,\mathrm{a}/2}=1.796$

The critical value is$1.796$

Part (b) Step 1: Given Information

It is given that $n=30$

Confidence Level$=0.95$

Part (b) Step 2: Calculation

Degree of freedom is calculated as:

$df=n-1=30-1=29$

Level of significance, $a=0.05$

EXCEL formula $=TINV(0.05,29)$gives

${\mathrm{t}}_{11,\mathrm{a}/2}=1.045$

The critical value is$1.045$

Part (c) Step 1: Given Information

It is given that $n=58$

Confidence Level$0.999$

Part (c) Step 2: Calculation

Degree of freedom is $df=n-1=58-1=57$

Level of Significance is $a=0.01$

EXCEL formula $=TINV(0.01,57)$gives

${\mathrm{t}}_{11,\mathrm{a}/2}=2.678$

The critical value is$2.678$