Construct Confidence Intervals for a Mean Response
-The data below are the scores of 10 randomly selected students from a statistics class and the number of hours they slept the night before the exam. Construct a $95 \%$ confidence interval about the mean value of $y$, the score on the final exam, given $x = 7$ hours, $\mathrm { y } = 5.044 \mathrm { x } + 56.11$ and $\mathrm { se } = 6.305$.
$\begin{array}{l|c|c|c|c|c|c|c|c|c|c|}
\text { Hours, } \mathrm{x} & 3 & 5 & 2 & 8 & 2 & 4 & 4 & 5 & 6 & 3 \
\hline \text { Scores, } \mathrm{y} & 65 & 80 & 60 & 88 & 66 & 78 & 85 & 90 & 90 & 71
\end{array}$

A) $( 82.840,99.996 )$
B) $( 74.54,108.30 )$
C) $( 77.21,110.45 )$
D) $( 79.16,112.34 )$

Question

Construct Confidence Intervals for a Mean Response
-The data below are the scores of 10 randomly selected students from a statistics class and the number of hours they slept the night before the exam. Construct a $95 \%$ confidence interval about the mean value of $y$, the score on the final exam, given $x = 7$ hours, $\mathrm { y } = 5.044 \mathrm { x } + 56.11$ and $\mathrm { se } = 6.305$.
$\begin{array}{l|c|c|c|c|c|c|c|c|c|c|}
\text { Hours, } \mathrm{x} & 3 & 5 & 2 & 8 & 2 & 4 & 4 & 5 & 6 & 3 \
\hline \text { Scores, } \mathrm{y} & 65 & 80 & 60 & 88 & 66 & 78 & 85 & 90 & 90 & 71
\end{array}$

A) $( 82.840,99.996 )$
B) $( 74.54,108.30 )$
C) $( 77.21,110.45 )$
D) $( 79.16,112.34 )$

Quizplus · Accepted Answer

Given the regression equation $y = 5.044x + 56.11$ and $x = 7$ hours, first calculate the mean value of $y$ for $x = 7$: $y = 5.044(7) + 56.11 = 91.418$. The confidence interval is calculated using the formula: $\text{mean} \pm (t_{\text{critical}} \times \text{SE})$. Without the critical value of $t$, we directly use the provided standard error ($\mathrm{se} = 6.305$) to estimate the range, assuming a 95% confidence level typically corresponds to a $t$ value that would give a margin of error in the range provided in option A, making it the closest estimate based on the given data and calculation.

Construct Confidence Intervals for a Mean Response -The Data Below 95%95 \%95%

Construct Confidence Intervals for a Mean Response
-The Data Below $95 \%$