Question

the data represent the heights in feet and the number of stories of the tallest buildings in a city.
height, x 947 708 658 529 450 446 430 420 419
stories, y 57 52 46 40 31 28 24 26 32
r ≈ 0.942
part: 0 / 3
part 1 of 3
find the equation of the regression line and draw the line on the scatter plot, but only if r is significant. round the slope and y - intercept to at least four decimal places, if necessary.
r is significant at the 5% level.
the equation of the regression line is y = + x.

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

Let \(n = 9\).
\(\bar{x}=\frac{947 + 708+658+529+450+446+430+420+419}{9}=\frac{4907}{9}\approx545.2222\)
\(\bar{y}=\frac{57 + 52+46+40+31+28+24+26+32}{9}=\frac{336}{9}\approx37.3333\)

Step2: Calculate the slope \(b_1\)

The formula for the slope \(b_1=r\frac{s_y}{s_x}\), where \(s_x=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}}\) and \(s_y=\sqrt{\frac{\sum_{i = 1}^{n}(y_i-\bar{y})^2}{n - 1}}\).
First, calculate \(\sum_{i=1}^{n}(x_i-\bar{x})^2\) and \(\sum_{i = 1}^{n}(y_i-\bar{y})^2\):
\(\sum_{i=1}^{n}(x_i-\bar{x})^2=(947 - 545.2222)^2+(708 - 545.2222)^2+\cdots+(419 - 545.2222)^2\)
\(\sum_{i=1}^{n}(y_i-\bar{y})^2=(57 - 37.3333)^2+(52 - 37.3333)^2+\cdots+(32 - 37.3333)^2\)
\(s_x=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{8}}\), \(s_y=\sqrt{\frac{\sum_{i = 1}^{n}(y_i-\bar{y})^2}{8}}\)
Since \(r = 0.942\), \(b_1=0.942\frac{s_y}{s_x}\approx0.0467\)

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

The formula for the y - intercept \(b_0=\bar{y}-b_1\bar{x}\)
\(b_0=37.3333-0.0467\times545.2222\)
\(b_0=37.3333 - 25.4619=11.8714\)

Answer:

\(y = 11.8714+0.0467x\)