Setp 1

Given:

\(\displaystyle\sigma={29}\)

\(\displaystyle{n}={54}\)

\(\displaystyle\overline{{{x}}}={97}\)

a) From the Standard Normal Table, the value of \(\displaystyle{z}\cdot\) for \(\displaystyle{95}\%\) level is 1.96

The \(\displaystyle{95}\%\) confidence interval for the population mean is obtained as below:

Sample statistic \(\displaystyle\pm{z}\cdot{S}{E}=\overline{{{x}}}\pm{z}\cdot{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)

\(\displaystyle={97}\pm{\left({1.96}\times{\frac{{{29}}}{{\sqrt{{{54}}}}}}\right)}\)

\(\displaystyle{97}\pm{7.7349}\)

\(\displaystyle={\left({89.2651},\ {104.7349}\right)}\)

Thus, the \(\displaystyle{95}\%\) confidence interval for the population mean is \(\displaystyle{\left({89.2651},\ {104.7349}\right)}.\)

Step 2

b) From the Standard Normal Table, the value of \(\displaystyle{z}\cdot\) for \(\displaystyle{90}\%\) level is 1.645.

The \(\displaystyle{90}\%\) confidence interval for the population mean is obtained as below:

Sample statistic \(\displaystyle\pm{z}\cdot{S}{E}=\overline{{{x}}}\pm{z}\cdot{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)

\(\displaystyle={97}\pm{\left({1.645}\times{\frac{{{29}}}{{\sqrt{{{54}}}}}}\right)}\)

\(\displaystyle={97}\pm{6.4918}\)

\(\displaystyle={\left({90.5082},\ {103.4918}\right)}\)

Thus, the \(\displaystyle{90}\%\) confidence interval for the population mean is \(\displaystyle{\left({90.5082},\ {103.4918}\right)}.\)

Step 3

c) From the Standard Normal Table, the value of \(\displaystyle{z}\cdot\) for \(\displaystyle{99}\%\) level is 2.576

The \(\displaystyle{99}\%\) confidence interval for the population mean is obtained as below:

Sample statistic \(\displaystyle\pm{z}\cdot{S}{E}=\overline{{{x}}}\pm{z}\cdot{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)

\(\displaystyle={97}\pm{\left({2.576}\times{\frac{{{29}}}{{\sqrt{{{54}}}}}}\right)}\)

\(\displaystyle={97}\pm{10.1659}\)

\(\displaystyle={\left({86.8341},\ {107.1659}\right)}\)

Thus, the \(\displaystyle{99}\%\) confidence interval for the population mean is \(\displaystyle{\left({86.8341},\ {107.1659}\right)}.\)

Given:

\(\displaystyle\sigma={29}\)

\(\displaystyle{n}={54}\)

\(\displaystyle\overline{{{x}}}={97}\)

a) From the Standard Normal Table, the value of \(\displaystyle{z}\cdot\) for \(\displaystyle{95}\%\) level is 1.96

The \(\displaystyle{95}\%\) confidence interval for the population mean is obtained as below:

Sample statistic \(\displaystyle\pm{z}\cdot{S}{E}=\overline{{{x}}}\pm{z}\cdot{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)

\(\displaystyle={97}\pm{\left({1.96}\times{\frac{{{29}}}{{\sqrt{{{54}}}}}}\right)}\)

\(\displaystyle{97}\pm{7.7349}\)

\(\displaystyle={\left({89.2651},\ {104.7349}\right)}\)

Thus, the \(\displaystyle{95}\%\) confidence interval for the population mean is \(\displaystyle{\left({89.2651},\ {104.7349}\right)}.\)

Step 2

b) From the Standard Normal Table, the value of \(\displaystyle{z}\cdot\) for \(\displaystyle{90}\%\) level is 1.645.

The \(\displaystyle{90}\%\) confidence interval for the population mean is obtained as below:

Sample statistic \(\displaystyle\pm{z}\cdot{S}{E}=\overline{{{x}}}\pm{z}\cdot{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)

\(\displaystyle={97}\pm{\left({1.645}\times{\frac{{{29}}}{{\sqrt{{{54}}}}}}\right)}\)

\(\displaystyle={97}\pm{6.4918}\)

\(\displaystyle={\left({90.5082},\ {103.4918}\right)}\)

Thus, the \(\displaystyle{90}\%\) confidence interval for the population mean is \(\displaystyle{\left({90.5082},\ {103.4918}\right)}.\)

Step 3

c) From the Standard Normal Table, the value of \(\displaystyle{z}\cdot\) for \(\displaystyle{99}\%\) level is 2.576

The \(\displaystyle{99}\%\) confidence interval for the population mean is obtained as below:

Sample statistic \(\displaystyle\pm{z}\cdot{S}{E}=\overline{{{x}}}\pm{z}\cdot{\frac{{\sigma}}{{\sqrt{{{n}}}}}}\)

\(\displaystyle={97}\pm{\left({2.576}\times{\frac{{{29}}}{{\sqrt{{{54}}}}}}\right)}\)

\(\displaystyle={97}\pm{10.1659}\)

\(\displaystyle={\left({86.8341},\ {107.1659}\right)}\)

Thus, the \(\displaystyle{99}\%\) confidence interval for the population mean is \(\displaystyle{\left({86.8341},\ {107.1659}\right)}.\)