# In a random sample of 25 people, the mean commute time to work was 34.1 minutes

In a random sample of 25 people, the mean commute time to work was 34.1 minutes and the standard deviation was 7.1 minutes. What is the margin of error of $$\displaystyle\mu$$?
Interpret the results.
A) With $$\displaystyle{80}\%$$ confidence, it can be said that the population mean commute time as between the bounds of the confidence interval.
B) With $$\displaystyle{80}\%$$ confidence, it can be said that the commute time is between the bounds of the confidence interval.
C) It can be said that $$\displaystyle{80}\%$$ of people have a commute time is between the bounds of the confidense interval
D) If a large sample of people are taken approximately $$\displaystyle{80}\%$$ of them will have commute times between the bounds of the confidence interval.

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unett

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.