Answer: [tex]11.7<\mu<15.9[/tex]
Step-by-step explanation:
Given : Significance level : [tex]\alpha:1-0.95=0.05[/tex]
Sample size : n=45
Critical value : [tex]z_{\alpha/2}=1.96[/tex]
Sample mean : [tex]\overline{x}=13.8\text{ years}[/tex]
Standard deviation : [tex]\sigma=7.3\text{ years}[/tex]
The confidence interval for population mean is given by :-
[tex]\overline{x}\pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}\\\\=13.8\pm(1.96)\dfrac{7.3}{\sqrt{45}}\\\\\approx13.8\pm2.1\\\\=(13.8-2.1,\ 13.8-2.1)=(11.7,\ 15.9)[/tex]
Hence, the 95% confidence interval estimate for the average number of years served by all Supreme Court justices is [tex]11.7<\mu<15.9[/tex]
