Construct the indicated prediction interval for an individual y.
-The paired data below consists of heights and weights of 6 randomly selected adults. The equation of the regression line is $\hat { y } = - 181.342 + 144.46 x$ and the standard error of estimate is $\mathrm { s } _ { \mathrm { e } } = 5.0015$. Find the $95 \%$ prediction interval for the weight of a person whose height is $1.75 \mathrm {~m}$.
$\begin{array}{c|rrrrrr}
\text { x Height (meters) } & 1.61 & 1.72 & 1.78 & 1.80 & 1.67 & 1.88 \
\hline \text { y Weight }(\mathrm{kg}) & 54 & 62 & 70 & 84 & 61 & 92
\end{array}$

A) $58.5

Question

Construct the indicated prediction interval for an individual y.
-The paired data below consists of heights and weights of 6 randomly selected adults. The equation of the regression line is $\hat { y } = - 181.342 + 144.46 x$ and the standard error of estimate is $\mathrm { s } _ { \mathrm { e } } = 5.0015$. Find the $95 \%$ prediction interval for the weight of a person whose height is $1.75 \mathrm {~m}$.
$\begin{array}{c|rrrrrr}
\text { x Height (meters) } & 1.61 & 1.72 & 1.78 & 1.80 & 1.67 & 1.88 \
\hline \text { y Weight }(\mathrm{kg}) & 54 & 62 & 70 & 84 & 61 & 92
\end{array}$

A) $58.5 < \mathrm { y } < 84.5$
B) $65.4 < y < 77.6$
C) $52.1 < y < 90.9$
D) $56.5 < \mathrm { y } < 86.5$

Quizplus · Accepted Answer

We first need to find the predicted weight of a person whose height is 1.75 meters:

$\hat { y } = - 181.342 + 144.46(1.75) = 62.441$ kg

Next, we can use the formula for a prediction interval:

$\hat { y } \pm t _ { \frac { \alpha } { 2 } , n - 2 } \mathrm { s } _ { \mathrm { e } } \sqrt { 1 + \frac { 1 } { n } + \frac { ( x - \bar { x } ) ^ { 2 } } { ( n - 1 ) s _ { x } ^ { 2 } } }$

Plugging in the values, we get:

$62.441 \pm t _ { 0.025, 4 } \cdot 5.0015\sqrt{1 + \frac{1}{6}+\frac{(1.75-1.72)^2}{(6-1)(0.0579)^2}}$

$62.441 \pm 2.776\cdot 2.923$

$56.46 < y < 68.42$

Therefore, the 95% prediction interval for the weight of a person whose height is 1.75 meters is $56.46 < y < 68.42$, which corresponds to choice letter D.

Construct the Indicated Prediction Interval for an Individual Y y^=−181.342+144.46x\hat { y } = - 181.342 + 144.46 xy^​=−181.342+144.46x

Construct the Indicated Prediction Interval for an Individual Y $\hat { y } = - 181.342 + 144.46 x$