Question

It’s critical Find the appropriate critical value for constructing a confidence interval in each of the following settings.

(a) Estimating a population proportion p at a $94\%$confidence level based on an SRS of size $125$.

(b) Estimating a population mean M at a $99\%$confidence level based on an SRS of size $58$.

Short answer

a). The critical value is $1.96$.

b). The critical value is $2.576$.

Step-by-step solution

Part (a) Step 1: Given Information

Population proportion $\left(p\right)=94\%=0.94$.

Sample size $\left(n\right)=125$.

Part (a) Step 2: Explanation

The level of significance is:

Level of significance $=1-$Confidence level

$=1-0.95$

$=0.05$

The critical value using standard normal table is calculated as:

$z_{\alpha /2}=z_{0.05/2}$

$=z_{0.025}$

$=1.96$

The critical value is $1.96$.

Part (b) Step 1: Given Information

Confidence level $=99\%$.

Sample size $\left(n\right)=58$.

Part (b) Step 2: Explanation

The level of significance is:

Level of significance $=1-$Confidence level

$=1-0.99$

$=0.01$

The critical value using the standard normal table is calculated as:

$z_{a/2}=z_{0.01/2}$

$=z_{0.005}$

$=2.576$