Use the given degree of confidence and sample data to construct a confidence interval for the population mean $\mu$ . Assume that the population has a normal distribution.
-$\mathrm { n } = 30 , \overline { \mathrm { x } } = 84.6 , \mathrm {~s} = 10.5,90 \%$ confidence
A) $79.32

Question

Use the given degree of confidence and sample data to construct a confidence interval for the population mean $\mu$ . Assume that the population has a normal distribution.
-$\mathrm { n } = 30 , \overline { \mathrm { x } } = 84.6 , \mathrm {~s} = 10.5,90 \%$ confidence
A) $79.32 < \mu < 89.88$
B) $81.36 < \mu < 87.84$
C) $80.68 < \mu < 88.52$
D) $81.34 < \mu < 87.86$

Quizplus · Accepted Answer

We use the formula $\mathrm { \overline { x } \pm t_{\frac{\alpha}{2},n-1} \times \frac { s } { \sqrt { n } } }$ where $t_{\frac{\alpha}{2},n-1}$ is the t-score with $n-1$ degrees of freedom and area $\frac{\alpha}{2}$ to find the confidence interval. By looking at the t-table with $n-1=29$ degrees of freedom and 90% confidence level, we find that $t_{0.05,29}=1.6991$. Plugging in the values from the problem, we get 
$$\mathrm { 84.6 \pm 1.6991 \times \frac { 10.5 } { \sqrt { 30 } } }$$
which simplifies to $81.34 < \mu < 87.86$. Therefore, the best choice is D.

Use the Given Degree of Confidence and Sample Data to Construct