Given the confidence interval formula as shown below.
[tex]CI=\bar{x}\pm t_{\frac{\alpha}{2}}\times\frac{s}{\sqrt{n}}[/tex][tex]\begin{gathered} \bar{x}\rightarrow mean \\ s\rightarrow standard\text{ deviation} \\ n\rightarrow sample\text{ size} \\ t_{\frac{\alpha}{2}}\rightarrow critical\text{ value} \end{gathered}[/tex]
The margin of error is
[tex]MOE=\pm t_{\frac{\alpha}{2}}\times\frac{s}{\sqrt{n}}[/tex]
Using an online calculator for the critical value
[tex]t_{\frac{\alpha}{2}}\text{ at 0.05 significance level is}\rightarrow\pm2.571[/tex]
The Margin of error MOE is given as:
[tex]\begin{gathered} MOE=\pm t_{\frac{\alpha}{2}}\times\frac{s}{\sqrt{n}} \\ \pm t_{\frac{\alpha}{2}}=\pm2.5706 \\ s=1,n=6 \\ \frac{s}{\sqrt{n}}=\frac{1}{\sqrt{6}}=0.4082 \\ MOE=\pm2.5706\times0.4082=\pm1.049 \end{gathered}[/tex]
Thus, the confidence interval is:
[tex]CI=\bar{x}\text{ }\pm MOE[/tex][tex]\begin{gathered} \bar{x}=8,MOE=\pm1.049 \\ CI=8\pm1.049 \\ CI\rightarrow(6.951,9.049) \end{gathered}[/tex]
