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. (a) x = 499 feet (b) x = 641 feet (c) x = 318 feet (d) x = 732 feet (a) predict the value of y for x = 499. choose the correct answer below. (b) predict the value of y for x = 641. choose the correct answer below. (c) predict the value of y for x = 318. choose the correct answer below.

Explanation:

First, we need to find the regression equation for the given data. Let's list the data points:
(768, 52), (628, 48), (518, 45), (511, 43), (491, 39), (478, 38)

Step 1: Calculate means of x and y

$\bar{x} = \frac{768+628+518+511+491+478}{6} = \frac{3394}{6} \approx 565.67$
$\bar{y} = \frac{52+48+45+43+39+38}{6} = \frac{265}{6} \approx 44.17$

Step 2: Calculate slope (b)

$b = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}$
Compute numerator:
$(768-565.67)(52-44.17) + (628-565.67)(48-44.17) + (518-565.67)(45-44.17) + (511-565.67)(43-44.17) + (491-565.67)(39-44.17) + (478-565.67)(38-44.17)$
$\approx (202.33)(7.83) + (62.33)(3.83) + (-47.67)(0.83) + (-54.67)(-1.17) + (-74.67)(-5.17) + (-87.67)(-6.17)$
$\approx 1584.24 + 238.72 - 39.57 + 63.96 + 386.04 + 541.92 \approx 2775.31$

Denominator:
$(202.33)^2 + (62.33)^2 + (-47.67)^2 + (-54.67)^2 + (-74.67)^2 + (-87.67)^2$
$\approx 40937.43 + 3885.03 + 2272.43 + 2988.81 + 5575.61 + 7686.03 \approx 63345.34$

$b \approx \frac{2775.31}{63345.34} \approx 0.0438$

Step 3: Calculate intercept (a)

$a = \bar{y} - b\bar{x} \approx 44.17 - 0.0438(565.67) \approx 44.17 - 24.78 \approx 19.39$

Regression equation: $\hat{y} = 19.39 + 0.0438x$

(a) Predict y when x=499

$\hat{y} = 19.39 + 0.0438(499) \approx 19.39 + 21.86 \approx 41.25 \approx 41$

Answer:

B. 41

(b) Predict y when x=641

$\hat{y} = 19.39 + 0.0438(641) \approx 19.39 + 28.08 \approx 47.47 \approx 48$