Question

Construct a $90\%$ confidence interval for the population mean time to complete the forms. State the confidence interval, sketch the graph and calculate the error bound.

Short answer

The confidence interval is$Cl:(7.9441,8.4559)$.

Step-by-step solution

Given Information

$90\%$ confidence interval for the population mean time to complete the forms.

Explanation

Let's calculate the Margin of error by using the given formula:

$\left[Z_{\frac{a}{2}}\times \frac{\sigma }{\sqrt{n}}\right]$

Place the values$$\begin{gathered} =\left[Z_{\frac{0.10}{2}}\times \frac{2.2}{\sqrt{200}}\right] \\ =\left[1.6449\times \frac{2.2}{\sqrt{200}}\right] \\ =0.25588 \end{gathered}$$

Then we have to find the confidence interval by using the given formula:

$\left[\overline{X}\pm Z_{\frac{a}{2}}\times \frac{\sigma }{\sqrt{n}}\right]$

role="math" localid="1652508772667" $$\begin{gathered} \text{Confidence Interval}=\left[8.2\pm Z_{\frac{0.10}{2}}\times \frac{2.2}{\sqrt{200}}\right] \\ =\left[8.2\pm 1.6449\times \frac{2.2}{\sqrt{200}}\right] \\ =[7.944,8.455] \end{gathered}$$

Therefore, the confidence interval $Cl:(7.9441,8.4559)$.

The graph:

The error bound $EBM=0.26$.