Step 1

\(\begin{array}{|c|c|c|}\hline & x & (x-\bar{x}) & (x-\bar{x})^{2} \\ \hline & 0.6 & -0.11429 & 0.01306122 \\ \hline & 0.82 & 0.10571 & 0.01117551 \\ \hline & 0.09 & -0.62429 & 0.38973265 \\ \hline & 0.89 & 0.17571 & 0.03087551 \\ \hline & 1.29 & 0.57571 & 0.33144694 \\ \hline & 0.49 & -0.22429 & 0.05030408 \\ \hline & 0.82 & 0.10571 & 0.01117551 \\ \hline Total & 5 & & 0.83777143 \\ \hline \end{array}\)

Here \(\displaystyle{n}={7}\)

Sampke mean \(\displaystyle=\overline{{{x}}}={\frac{{\sum{x}}}{{{n}}}}={0.714}\)

Sample \(\displaystyle{S}.{D}.={s}=\sqrt{{{\frac{{{1}}}{{{n}-{1}}}}\sum{\left({x}-\overline{{{x}}}\right)}^{{{2}}}}}={0.3737}\)

Step 2

\(\begin{array}{|c|}\hline \text{Confidence Level}=95 \\ \hline \text{Significance Level}=\alpha=(100-95)\%=0.05 \\ \hline \text{Degrees of freedom}=n-1=7-1=6 \\ \hline \text{Critical value}=t^{*}=2.447 [\text{using Excel}=TINV(0.05,\ 6)]\\ \hline \end{array}\)

Standard Error \(\displaystyle={\frac{{{s}}}{{\sqrt{{{n}}}}}}={0.1412}\)

Margin of Error (M.E) \(\displaystyle={t}^{{\cdot}}{\frac{{{s}}}{{\sqrt{{{n}}}}}}={0.3456}\)

\(\displaystyle\text{Lower Limit}=\overline{{{x}}}-{\left({M}.{E}\right)}={0.3687}\)

\(\displaystyle\text{Upper Limit}=\overline{{{x}}}+{\left({M}.{E}\right)}={1.0599}\)