The regression equation for the given paired data is $\hat { y } = 73.012 + 0.8859 \mathrm { x }$ and the standard error of estimate is $\mathrm { s } _ { \mathrm { e } } = 2.78807$. Find the $95 \%$ prediction interval of $\mathrm { y }$ for $\mathrm { x } = 37$.
$\begin{array}{r|rrrrrr}
\mathrm{x} & 25 & 26 & 36 & 36 & 40 & 48 \
\hline \mathrm{y} & 95 & 95 & 102 & 109 & 110 & 114
\end{array}$
A) $97.4

Question

The regression equation for the given paired data is $\hat { y } = 73.012 + 0.8859 \mathrm { x }$ and the standard error of estimate is $\mathrm { s } _ { \mathrm { e } } = 2.78807$. Find the $95 \%$ prediction interval of $\mathrm { y }$ for $\mathrm { x } = 37$.
$\begin{array}{r|rrrrrr}
\mathrm{x} & 25 & 26 & 36 & 36 & 40 & 48 \
\hline \mathrm{y} & 95 & 95 & 102 & 109 & 110 & 114
\end{array}$
A) $97.4 < y < 10.8$
B) $73.0 < y < 114.2$
C) $73.0 < y < 105.8$
D) $97.4 <$ y $< 114.2$

Quizplus · Accepted Answer

To find the 95% prediction interval for $y$ when $x = 37$, we first calculate the predicted value of $y$ using the given regression equation: $\hat { y } = 73.012 + 0.8859(37) = 105.7923$. The prediction interval is calculated as $\hat { y } \pm t_{\alpha/2} \cdot s_e$, where $t_{\alpha/2}$ is the t-value for the desired confidence level (not provided, but typically found in a t-distribution table and depends on the degrees of freedom, which is related to the sample size), and $s_e$ is the standard error of estimate. Without the exact t-value, we can't calculate the exact bounds, but the correct interval must be centered around the predicted value of 105.7923. Only option D is centered around the predicted value and makes sense in the context of the given data and calculation.

The Regression Equation for the Given Paired Data Is y^=73.012+0.8859x\hat { y } = 73.012 + 0.8859 \mathrm { x }y^​=73.012+0.8859x

The Regression Equation for the Given Paired Data Is $\hat { y } = 73.012 + 0.8859 \mathrm { x }$