Step 1

Given,

sample size \(\displaystyle{\left({n}\right)}={36}\)

sample mean \(\displaystyle{\left(\overline{{{x}}}\right)}={46}\)

standard deviation \(\displaystyle{\left(\sigma\right)}={7.4}\)

\(\displaystyle\alpha={1}-{0.90}={0.1}\)

\(\displaystyle{\frac{{\alpha}}{{{2}}}}={0.05}\)

\(\displaystyle{Z}_{{{0.05}}}={1.645}\)

Step 2

\(\displaystyle{90}\%\) confidence interval \(\displaystyle{\left({C}.{I}.\right)}:\overline{{{X}}}\pm{Z}_{{{\frac{{\alpha}}{{{2}}}}}}\times{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)

\(\displaystyle{C}.{I}={46}\pm{1.645}\times{\frac{{{7.4}}}{{\sqrt{{{36}}}}}}\)

\(\displaystyle{C}.{I}={46}\pm{2.029}\)

\(\displaystyle{C}.{I}={\left({46}-{2.029},{46}+{2.029}\right)}\)

\(\displaystyle{C}.{I}={\left({43.971},{48.029}\right)}\)

\(\displaystyle{C}.{I}\stackrel{\sim}{=}{\left({44.0},{48.0}\right)}\)