Step 1
a.
It is given that, Sample mean = 108.
Population standard deviation, \(\displaystyle\sigma={26}\).
The sample size, n is 43.
The z-critical value for 95% confidence interval is 1.96.
The 95% confidence interval can be calculated as follows:
\(\displaystyle{C}{I}=\overline{{{x}}}\pm{z}{\left({\frac{{\sigma}}{{\sqrt{{{n}}}}}}\right)}\)

\(\displaystyle={108}\pm{1.96}{\left({\frac{{{26}}}{{\sqrt{{{43}}}}}}\right)}\)

\(\displaystyle={108}\pm{7.7713}\)

\(\displaystyle={\left({100.2287},{115.7713}\right)}\) The 95% confidence interval is (100.2287, 115.7713). Step 2 b. Population standard deviation, \sigma is 58. The 95% confidence interval can be calculated as follows: \(\displaystyle{C}{I}=\overline{{{x}}}\pm{z}{\left({\frac{{\sigma}}{{\sqrt{{{n}}}}}}\right)}\)

\(\displaystyle={108}\pm{1.96}{\left({\frac{{{58}}}{{\sqrt{{{43}}}}}}\right)}\)

\(\displaystyle={108}\pm{17.3360}\)

\(\displaystyle={\left({90.664},{125.336}\right)}\) The 95% confidence interval is (90.664,125.336). Step 3 c. Population standard deviation,SPK \sigma=8ZSK. The 95% confidence interval can be calculated as follows: \(\displaystyle{C}{I}=\overline{{{x}}}\pm{z}{\left({\frac{{\sigma}}{{\sqrt{{{n}}}}}}\right)}\)

\(\displaystyle={108}\pm{1.96}{\left({\frac{{{8}}}{{\sqrt{{{43}}}}}}\right)}\)

\(\displaystyle={108}\pm{2.3911}\)

\(\displaystyle={\left({105.6089},{110.3911}\right)}\) The 95% confidence interval is (105.6089,110.3911).

\(\displaystyle={108}\pm{1.96}{\left({\frac{{{26}}}{{\sqrt{{{43}}}}}}\right)}\)

\(\displaystyle={108}\pm{7.7713}\)

\(\displaystyle={\left({100.2287},{115.7713}\right)}\) The 95% confidence interval is (100.2287, 115.7713). Step 2 b. Population standard deviation, \sigma is 58. The 95% confidence interval can be calculated as follows: \(\displaystyle{C}{I}=\overline{{{x}}}\pm{z}{\left({\frac{{\sigma}}{{\sqrt{{{n}}}}}}\right)}\)

\(\displaystyle={108}\pm{1.96}{\left({\frac{{{58}}}{{\sqrt{{{43}}}}}}\right)}\)

\(\displaystyle={108}\pm{17.3360}\)

\(\displaystyle={\left({90.664},{125.336}\right)}\) The 95% confidence interval is (90.664,125.336). Step 3 c. Population standard deviation,SPK \sigma=8ZSK. The 95% confidence interval can be calculated as follows: \(\displaystyle{C}{I}=\overline{{{x}}}\pm{z}{\left({\frac{{\sigma}}{{\sqrt{{{n}}}}}}\right)}\)

\(\displaystyle={108}\pm{1.96}{\left({\frac{{{8}}}{{\sqrt{{{43}}}}}}\right)}\)

\(\displaystyle={108}\pm{2.3911}\)

\(\displaystyle={\left({105.6089},{110.3911}\right)}\) The 95% confidence interval is (105.6089,110.3911).