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.
-The football coach randomly selected ten players and timed how long each player took to perform a certain drill. The times (in minutes) were: $\begin{array} { l l l l l } 7.0 & 10.8 & 9.5 & 8.0 & 11.5 \end{array}$
$\begin{array} { l l l l l } 7.5 & 6.4 & 11.3 & 10.2 & 12.6 \end{array}$
Determine a $95 \%$ confidence interval for the mean time for all players.
A) $10.93 \mathrm {~min}

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.
-The football coach randomly selected ten players and timed how long each player took to perform a certain drill. The times (in minutes) were: $\begin{array} { l l l l l } 7.0 & 10.8 & 9.5 & 8.0 & 11.5 \end{array}$
$\begin{array} { l l l l l } 7.5 & 6.4 & 11.3 & 10.2 & 12.6 \end{array}$
Determine a $95 \%$ confidence interval for the mean time for all players.
A) $10.93 \mathrm {~min} < \mu < 8.03 \mathrm {~min}$
B) $11.03 \mathrm {~min} < \mu < 7.93 \mathrm {~min}$
C) $7.93 \mathrm {~min} < \mu < 11.03 \mathrm {~min}$
D) $8.03 \mathrm {~min} < \mu < 10.93 \mathrm {~min}$

Quizplus · Accepted Answer

We will use the t-distribution since the population standard deviation is unknown and the sample size is small (n=10). 
Using a t-interval with a 95% confidence level and 9 degrees of freedom (df = n-1), we find the t-value to be 2.262.
The sample mean is: 
\begin{align*}
\bar{x} &= \frac{7.0+10.8+9.5+8.0+11.5+7.5+6.4+11.3+10.2+12.6}{10} \
&= 9.13
\end{align*}
The standard error is:
\begin{align*}
SE &= \frac{s}{\sqrt{n}}\
&= \frac{2.349}{\sqrt{10}}\
&= 0.743
\end{align*}
The confidence interval is then:
\begin{align*}
\bar{x} \pm t^* 	imes SE &= 9.13 \pm 2.262 	imes 0.743 \
&= 9.13 \pm 1.677\
&= (7.453, 10.807)
\end{align*}
Therefore, the best choice is D.

Use the Given Degree of Confidence and Sample Data to Construct