Step 1

From the provided information,

Population standard deviation \(\displaystyle{\left(\sigma\right)}={14.8}\)

Sample size \(\displaystyle{\left({n}\right)}={21}\)

Sample mean \(\displaystyle{\left(\overline{{{x}}}\right)}={139.05}\)

a) Since, population standard deviation is known therefore z distribution will be used.

The z value at \(\displaystyle{99}\%\) confidence level from the standard normal table is 2.58.

The required \(\displaystyle{99}\%\) confidence interval for µ can be obtained as:

\(\displaystyle{C}{I}=\overline{{{x}}}\pm{z}_{{\frac{\alpha}{{2}}}}{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)

\(\displaystyle={139.05}\pm{\left({2.58}\right)}{\frac{{{14.8}}}{{\sqrt{{{21}}}}}}\)

\(\displaystyle={139.05}\pm{8.33}\)

\(\displaystyle={\left({130.72},\ {147.38}\right)}\)

Thus, the lower bound of \(\displaystyle{99}\%\) confidence interval is \(\displaystyle{130.72}\) and the upper bound is 147.38.

Step 2

b) The z value at \(\displaystyle{97}\%\) confidence level from the standard normal table is 2.17.

The required \(\displaystyle{97}\%\) confidence interval for \(\displaystyle\mu\) can be obtained as:

\(CI=\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\)

\(\displaystyle{139.05}\pm{\left({2.17}\right)}{\frac{{{14.8}}}{{\sqrt{{{21}}}}}}\)

\(\displaystyle={139.50}\pm{7.01}\)

\(\displaystyle{\left({132.04},\ {146.06}\right)}\)

Thus, the lower bound of \(\displaystyle{97}\%\) confidence interval is 132.04 an the upper bound is 146.06

Step 3

c) The z value at \(\displaystyle{95}\%\) confidence level from the standard normal table is 1.96.

The required \(\displaystyle{95}\%\) confidence interval for \(\displaystyle\mu\) can be obtained as:

\(\displaystyle{C}{I}=\overline{{{x}}}\pm{z}_{{\frac{\alpha}{{2}}}}{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)

\(\displaystyle{139.05}\pm{\left({1.96}\right)}{\frac{{{14.8}}}{{\sqrt{{{21}}}}}}\)

\(\displaystyle={139.05}\pm{6.33}\)

\(\displaystyle={\left({132.72},\ {145.38}\right)}\)

Thus, the lower bound of \(\displaystyle{95}\%\) confidence interval is 132.72 and the upper bound is 145.38