Question

find the equation of the regression line for the given data. then construct a scatter plot of the data and draw the regression line. (the pair of variables have a significant correlation.) then use the regression equation to predict the value of y for each of the given x - values, if meaningful. the table below shows the heights (in feet) and the number of stories of six notable buildings in a city.\
height, x\
stories, y\
768\
52\
628\
48\
518\
45\
511\
43\
491\
39\
478\
38\
(a) x = 499 feet\
(b) predict the value of y for x = 641. choose the correct answer below.\
a. 41\
b. 48\
c. 33\
d. not meaningful\
(c) predict the value of y for x = 318. choose the correct answer below.\
a. 33\
b. 48\
c. 33\
d. not meaningful\
(d) predict the value of y for x = 732. choose the correct answer below.\
a. 41\
b. 52\
c. 33\
d. not meaningful

Explanation:

First, we calculate the regression line equation for the given data (x: height, y: stories):
Data points: (499,38), (491,39), (511,43), (518,45), (628,48), (768,52)

Step1: Compute means

$\bar{x} = \frac{499+491+511+518+628+768}{6} = \frac{3415}{6} \approx 569.17$
$\bar{y} = \frac{38+39+43+45+48+52}{6} = \frac{265}{6} \approx 44.17$

Step2: Calculate slope (b)

Using formula $b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}$
After calculation, $b \approx 0.0426$

Step3: Calculate intercept (a)

$a = \bar{y} - b\bar{x} \approx 44.17 - 0.0426(569.17) \approx 19.8$

Regression equation: $\hat{y} = 19.8 + 0.0426x$

Predictions:

(a) $x=499$: $\hat{y} \approx 19.8 + 0.0426(499) \approx 39.1 \approx 39$ (closest to 38, but options not shown—assuming context aligns with given choices for other parts)
(b) $x=641$: $\hat{y} \approx 19.8 + 0.0426(641) \approx 19.8 + 27.3 \approx 47.1 \approx 48$ → B
(c) $x=318$: Outside the range of x-values (491-768) → extrapolation, not meaningful → D
(d) $x=732$: Outside the range of x-values → extrapolation, not meaningful → D

Answer:

(b) B. 48
(c) D. not meaningful
(d) D. not meaningful

(Note: For part (a), since options are not provided in the visible prompt, we focus on the parts with clear options as per the image.)