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 μ at a 99% confidence level based on an SRS of size 58

Short answer

Part a) The critical value is$1.88$

Part b) The critical value is$2.665$

Step-by-step solution

Part a) Step 1: Given information

Confidence level$=94\%$

Sample size (n)$=125$

Part a) Step 2: Calculation

Level of significance$\left(\alpha \right)=1-0.94=0.06$

Using the standard normal, the critical value is set at$6\%$

level of significance can be calculated as:

$$\begin{gathered} z_{\frac{\alpha }{2}}=z_{\frac{0.06}{2}} \\ =1.88 \end{gathered}$$

Therefore, the critical value is$1.88$

Part b) Step 1: Given information

Confidence level$=99\%$

Sample size (n)$=58$

Part b) Step 2: Calculation

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

When the sample standard deviation is known, the $t$-critical value must be calculated.

The degree of freedom can be calculated as:

$$\begin{gathered} df=n-1 \\ =58-1 \\ =57 \end{gathered}$$

The critical value at the $1\%$level of significance is calculated as follows:

$$\begin{gathered} t_{\frac{\alpha }{2},df}=t_{\frac{0.01}{2},57} \\ =2.665 \end{gathered}$$

Therefore, the critical value is$2.665$