Step 1

The mean and standard deviation are obtained as follows:

\(\begin{array}{|c|c|} \hline &x&x^{2} \\ \hline & 19.17&367.4889 \\ \hline & 21.18&448.5924\\ \hline &20.38&415.3444\\ \hline &25.08&629.0064\\ \hline SUM&85.81&1860.432\\ \hline \end{array}\)

Mean \(\displaystyle={\left(\overline{{{x}}}\right)}={\frac{{\sum{x}}}{{{n}}}}={\frac{{{85.81}}}{{{4}}}}={21.45}\)

Standard Deviation \(\displaystyle=\sqrt{{{\frac{{\sum{x}^{{{2}}}-{\frac{{{\left(\sum{x}\right)}}}{{{n}}}}}}{{{n}-{1}}}}}}\)

Standard Deviation \(\displaystyle=\sqrt{{{\frac{{{1860.432}-{\frac{{{85.81}^{{{2}}}}}{{{4}}}}}}{{{3}}}}}}\)

Standard Deviation \(\displaystyle={2.55}\)

Step 2

The \(\displaystyle{90}\%\) confidence interval is obtained as follows:

Confidence Interval \(\displaystyle={\left[\overline{{{x}}}\pm{t}_{{{\left({n}-{1},{\frac{{\alpha}}{{{2}}}}\right)}}}\times{\frac{{{s}}}{{\sqrt{{{n}}}}}}\right]}\)

Confidence Intarval \(\displaystyle={\left[{21.45}\pm{t}_{{{\left({3},{\frac{{{0.10}}}{{{2}}}}\right)}}}\times{\frac{{{2.55}}}{{\sqrt{{{4}}}}}}\right]}\)

Confidence Interval \(\displaystyle={\left[{21.45}\pm{2.353}\times{\frac{{{2.55}}}{{\sqrt{{{4}}}}}}\right]}\)

Comfidence Interval \(\displaystyle={\left[{18.45},{24.45}\right]}\)

Lower Limit \(\displaystyle={18.45}\)

Upper Limit \(\displaystyle={24.45}\)