91Ó°ÊÓ

$\stackrel{¯}{x}=30,n=25,\mathrm{σ}=4$

The formula used:$\stackrel{¯}{x}Â\pm 2\frac{\mathrm{σ}}{\sqrt{n}}$

Compute a $95\%$confidence interval for the population mean.

Consider $\stackrel{¯}{x}=30,n=25$, and $\mathrm{σ}=4$

Empirical rule:

Property 1: Around $68\%$of the data set is located between $(\stackrel{¯}{x}-s,\stackrel{¯}{x}+s)$

Property 2: Approximately $95\%$of the data set is located between $(\stackrel{¯}{x}-2s,\stackrel{¯}{x}+2s)$

Property 3: Approximately $99.7\%$of the data set is located between $(\stackrel{¯}{x}-3s,\stackrel{¯}{x}+3s)$

By using Property $2$, the $95\%$of all observations lie within two standard deviations to either side of the mean.

The $95\%$confidence interval for the population mean is,

$$\begin{gathered} \stackrel{¯}{x}Â\pm 2\frac{\mathrm{σ}}{\sqrt{n}}=30Â\pm 2\left(\frac{4}{\sqrt{25}}\right) \\ =30Â\pm 1.6 \\ =(30-1.6,30+1.6) \\ =(28.4,31.6) \end{gathered}$$

Thus, the confidence interval for the population mean is $(28.4,31.6)$

From part (a), the margin of error is $1.6$

With $95\%$ confidence, the population means $\left(\mathrm{μ}\right)$ may be predicted to within $1.6$

The confidence interval's endpoints should be expressed.

Thus, the endpoints of the confidence interval is $30Â\pm 1.6$