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

ddaeeric 2021-09-27 Answered

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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Expert Answer

unett
Answered 2021-09-28 Author has 11464 answers

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.

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Interpret the results.
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