Question

the table below shows the house price (in 1,000s) and the square feet of each house. calculate the correlation coefficient and regression equation for the data. round values to the nearest hundredth. (type your answer with no spaces.) r≈0.7621 y =

Explanation:

Step1: Calculate means of x and y

Let \(x_i\) be the square - feet values and \(y_i\) be the house - price values.
\(n = 10\)
\(\bar{x}=\frac{1400 + 1600+1700+1875+1100+1550+2350+2450+1425+1700}{10}=\frac{17150}{10}=1715\)
\(\bar{y}=\frac{245+312+279+308+199+219+405+324+319+255}{10}=\frac{2865}{10}=286.5\)

Step2: Calculate numerator and denominator for correlation coefficient

The formula for the correlation coefficient \(r\) is \(r=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_i-\bar{x})^2\sum_{i = 1}^{n}(y_i-\bar{y})^2}}\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})=(1400 - 1715)(245-286.5)+(1600 - 1715)(312 - 286.5)+\cdots+(1700 - 1715)(255 - 286.5)\)
\(=( - 315)(-41.5)+(-115)(25.5)+\cdots+(-15)(-31.5)\)
\(=13072.5-2932.5+\cdots + 472.5\)
\(\sum_{i = 1}^{n}(x_i-\bar{x})^2=(1400 - 1715)^2+(1600 - 1715)^2+\cdots+(1700 - 1715)^2\)
\(=(-315)^2+(-115)^2+\cdots+(-15)^2\)
\(\sum_{i = 1}^{n}(y_i-\bar{y})^2=(245 - 286.5)^2+(312 - 286.5)^2+\cdots+(255 - 286.5)^2\)
After calculating these sums:
\(r\approx0.7621\)
For the regression equation \(y = a+bx\), where \(b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\) and \(a=\bar{y}-b\bar{x}\)
\(b=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\approx0.14\)
\(a = 286.5-0.14\times1715=286.5 - 240.1=46.4\)
The regression equation is \(y = 46.4+0.14x\)

Answer:

\(r\approx0.76\), \(y = 46.40+0.14x\)