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	774	625	521	508	497	477
stories, y	51	47	45	42	38	35

(a) x = 498 feet
(b) x = 630 feet
(c) x = 315 feet
(d) x = 728 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 \(n = 6\).
We need to calculate \(\sum x\), \(\sum y\), \(\sum xy\), \(\sum x^{2}\).
\(\sum x=774 + 626+521+508+497+477 = 3403\)
\(\sum y=51 + 47+45+42+38+35 = 258\)
\(\sum xy=(774\times51)+(626\times47)+(521\times45)+(508\times42)+(497\times38)+(477\times35)\)
\(=39474+29422+23445+21336+18886+16695 = 149258\)
\(\sum x^{2}=774^{2}+626^{2}+521^{2}+508^{2}+497^{2}+477^{2}\)
\(=599076+391876+271441+258064+247009+227529 = 1994995\)

Step2: Calculate the slope \(b_1\)

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

$$\begin{align*} b_1&=\frac{6\times149258 - 3403\times258}{6\times1994995-(3403)^{2}}\\ &=\frac{895548-877974}{11969970 - 11580409}\\ &=\frac{17574}{389561}\\ &\approx0.045 \end{align*}$$

\]

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

The formula for the y - intercept \(b_0=\bar{y}-b_1\bar{x}\), where \(\bar{x}=\frac{\sum x}{n}=\frac{3403}{6}\approx567.17\) and \(\bar{y}=\frac{\sum y}{n}=\frac{258}{6} = 43\)
\[

$$\begin{align*} b_0&=43-0.045\times567.17\\ &=43 - 25.52265\\ &\approx17.48 \end{align*}$$

\]

Answer:

\(\hat{y}=0.045x + 17.48\)