91Ó°ÊÓ

$\stackrel{¯}{x}=30,n=25,\mathrm{σ}=4$, confidence level $=90\%$

The formula used: the confidence interval$\stackrel{¯}{x}Â\pm z_{\frac{a}{2}}\left(\frac{\mathrm{σ}}{\sqrt{n}}\right)$and$\text{Margin of error}\left(E\right)=z_{\frac{a}{2}}\left(\frac{\mathrm{σ}}{\sqrt{n}}\right)$

Compute the $90\%$confidence interval for $\mu$.

The needed value of $z_{\frac{\mathrm{Î\pm }}{2}}$ with a $90\%$confidence level is$1.645$ according to "Table II Areas under the standard normal curve."

Thus, the confidence interval is,

$$\begin{gathered} \stackrel{¯}{x}Â\pm z_{\frac{a}{2}}\left(\frac{\mathrm{σ}}{\sqrt{n}}\right)=30Â\pm 1.645\left(\frac{4}{\sqrt{25}}\right) \\ =30Â\pm 1.645(0.8) \\ =30Â\pm 1.316 \\ =(28.684,31.316) \end{gathered}$$

Therefore, the $90\%$confidence interval for $\mathrm{μ}$is $(28.684,31.316)$.

Using the half-length of the confidence interval, calculate the margin of error.
$$\begin{gathered} \text{Margin of error}=\frac{\text{Upper limit}-\text{Lower limit}}{2} \\ =\frac{31.315-28.684}{2} \\ =\frac{2.631}{2} \\ =1.316 \end{gathered}$$

Thus, the margin of error by using the half-length of the confidence interval is $1.316$

Using a formula, calculate the margin of error.

Thus, the margin of error by using formula is $1.316$