Step 1

The \(\displaystyle{\left({1}-\alpha\right)}{100}\%\) confidence interval formula for the population mean when population standard deviation is not known, is defined as follows:

\(\displaystyle{C}{I}=\overline{{{x}}}\pm{t}_{{{\frac{{\alpha}}{{{2}}}},\ {n}-{1}}}{\left({\frac{{{s}}}{{\sqrt{{{n}}}}}}\right).}\)

Here, \(\displaystyle{t}_{{\frac{\alpha}{{2}},\ {n}-{1}}}\) is the critical value of the t-distribution with degrees of freedom of \(\displaystyle{n}-{1}\) above which, \(\displaystyle{100}{\left(\frac{\alpha}{{2}}\right)}\%\) or \(\displaystyle{\left({1}-\frac{\alpha}{{2}}\right)}\) proportion of the observation of the observations lie, and below which, \(\displaystyle{100}{\left({1}-\alpha+\frac{\alpha}{{2}}\right)}\%={100}{\left({1}-\frac{\alpha}{{2}}\right)}\%\) or \(\displaystyle{\left({1}-\frac{\alpha}{{2}}\right)}\) proportion of the observations lie, \(\displaystyle\overline{{{x}}}\) is the sample mean, s is the sample standard deviation, and n is the sample size.

Step 2

The sample size \(\displaystyle{n}={26}\)

The sample mean is and the sample standard deviation is \(\displaystyle{s}={7.1}\)

The degrees of freedom is \(\displaystyle{25}={\left({26}-{1}\right)}\)

The confidence level is 0.80. Hence, the level of significance is \(\displaystyle{1}-{0.80}={0.20}\)

Using Excel formula: =T.INV.2T(0.20,25), the critical value is, \(\displaystyle{t}_{{\frac{\alpha}{{2}}}}={t}_{{{0.10}}}\approx{1.316}\)

Thus,

\(\displaystyle{C}{I}=\overline{{{x}}}\pm{t}_{{{\frac{{\alpha}}{{{2}}}},\ {n}-{1}}}{\left({\frac{{{s}}}{{\sqrt{{{n}}}}}}\right)}\)

\(\displaystyle={34.1}\pm{\left({1.316}\right)}{\left({\frac{{{7.1}}}{{\sqrt{{{26}}}}}}\right)}\)

\(\displaystyle={34.1}\pm{1.83243072}\)

\(\displaystyle={\left({32.3},\ {35.9}\right)}\)

Thus, the \(\displaystyle{80}\%\) confidence interval for the population mean \(\displaystyle\mu\) is (32.3 minutes, 35.9 minutes).

The margin of error of \(\displaystyle\mu\) is,

\(\displaystyle{t}_{{{\frac{{\alpha}}{{{2}}}},\ {n}-{1}}}{\left({\frac{{{s}}}{{\sqrt{{{n}}}}}}\right)}={\left({1.316}\right)}{\left({\frac{{{7.1}}}{{\sqrt{{{26}}}}}}\right)}\)

\(\displaystyle={1.8}\)

It can be said with \(\displaystyle{80}\%\) confidence that, the population mean commute time is between the bounds of the confidence interval.

Correct option is A.