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	772	628	518	508	496	483
stories, y	51	48	46	41	38	37

(a) x = 501 feet
(b) x = 641 feet
(c) x = 315 feet
(d) x = 726 feet
find the regression equation
\\(\hat{y}=\square x+\square\\)
(round the slope to three decimal places as needed. round the y - intercept to two decimal places as needed.)

Explanation:

Step1: Calculate necessary sums

Let \(n = 6\).
Calculate \(\sum x=772 + 628+518+508+496+483=3405\), \(\sum y=51 + 48+46+41+38+37 = 261\), \(\sum xy=772\times51+628\times48 + 518\times46+508\times41+496\times38+483\times37=772\times51+628\times48+518\times46+508\times41+496\times38+483\times37 = 147793\), \(\sum x^{2}=772^{2}+628^{2}+518^{2}+508^{2}+496^{2}+483^{2}=1979947\).

Step2: Calculate the slope \(b_1\)

The formula for the slope \(b_1=\frac{n\sum xy-\sum x\sum y}{n\sum x^{2}-(\sum x)^{2}}\).
\[

$$\begin{align*} b_1&=\frac{6\times147793 - 3405\times261}{6\times1979947-3405^{2}}\\ &=\frac{886758-888705}{11879682 - 11594025}\\ &=\frac{-1947}{285657}\approx - 0.007 \end{align*}$$

\]

Step3: Calculate the y - intercept \(b_0\)

The formula for the y - intercept \(b_0=\bar{y}-b_1\bar{x}\), where \(\bar{x}=\frac{\sum x}{n}=\frac{3405}{6}=567.5\) and \(\bar{y}=\frac{\sum y}{n}=\frac{261}{6}=43.5\).
\[

$$\begin{align*} b_0&=43.5-(-0.007)\times567.5\\ &=43.5 + 3.9725\\ &\approx47.47 \end{align*}$$

\]
The regression equation is \(\hat{y}=-0.007x + 47.47\)

Step4: Predict values of y

(a) When \(x = 501\)

\(\hat{y}=-0.007\times501+47.47=-3.507 + 47.47=43.963\approx44\)

(b) When \(x = 641\)

\(\hat{y}=-0.007\times641+47.47=-4.487+47.47 = 42.983\approx43\)

(c) When \(x = 315\)

\(\hat{y}=-0.007\times315+47.47=-2.205 + 47.47=45.265\approx45\)

(d) When \(x = 726\)

\(\hat{y}=-0.007\times726+47.47=-5.082+47.47 = 42.388\approx42\)

Answer:

The regression equation is \(\hat{y}=-0.007x + 47.47\)
(a) 44
(b) 43
(c) 45
(d) 42