Solve the Problem $y = \beta _ { 0 } + \beta _ { 1 } x$

Multiple Choice

Solve the problem.
-After obtaining a regression line $y = \beta _ { 0 } + \beta _ { 1 } x$ , a confidence interval for the mean of all values $y$ for which $x = x 0$ can be obtained as follows:
$( \hat { y } - \mathrm { E } , \hat { \mathrm { y } } + \mathrm { E } )$ where $E = \left( t _ { \alpha / 2 } \right) \operatorname { se } \sqrt { \frac { 1 } { n } + \frac { n \left( x _ { 0 } - \bar { x } \right) ^ { 2 } } { n \sum x ^ { 2 } - \left( \sum x \right) ^ { 2 } } }$
The critical value $t _ { \alpha / 2 }$ is found from the t-table using $n - 2$ degrees of freedom.
Use the data below to obtain a $95 \%$ confidence interval estimate of the mean test score of all students who study $10.5$ hours. Note that the equation of the regression line is $\hat { y } = 52.8804 + 3.405 \mathrm { x }$ and that $\mathrm { s } = 6.8359$ .
$\begin{array}{l|lllll}x \text { (hours studied) } & 2.5 & 4.5 & 5.1 & 7.9 & 11.6 \\\hline \mathrm{y} \text { (score on test) } & 66 & 70 & 60 & 83 & 93\end{array}$

A) $( 80.48,96.78 )$
B) $( 77.11,100.15 )$
C) $( 72.49,104.78 )$
D) $( 61.55,115.71 )$