Answer:
[tex]\displaystyle y=-0.927x+13.63[/tex]
Step-by-step explanation:
Simple Linear Regression
It a function that represents the relationship between two or more variables in a given data set. It uses the method of the least-squares regression line which minimizes the error between the estimate function and the real data.
Let's compute the best-fit line for the data
[tex]x=\{1,5,6,12,15\}[/tex]
[tex]y=\{14,11,4,2,1\}[/tex]
First, we find the sums
[tex]\displaystyle \sum x=1+5+6+12+15=39[/tex]
[tex]\displaystyle \sum y=14+11+4+2+1=32[/tex]
Then, we compute the averages values
[tex]\displaystyle \bar{x}=\frac{39}{5}=7.8[/tex]
[tex]\displaystyle \bar{y}=\frac{32}{5}=6.4[/tex]
We will also compute the sums of the cross-products and the sum of the squares
[tex]\displaystyle \sum xy=(1)(14)+(5)(11)+(6)(4)+(12)(2)+(15)(1)=137[/tex]
[tex]\displaystyle \sum x^2=1^2+5^2+6^2+12^2+15^2=1+25+36+144+225[/tex]
[tex]\displaystyle \sum x^2=431[/tex]
We will compute Sxy and Sxx
[tex]\displaystyle S_{xy}=\sum xy-\frac{\sum x\ \sum y}{n}[/tex]
[tex]\displaystyle S_{xy}=137-\frac{(39)(32)}{5}[/tex]
[tex]\displaystyle S_{xy}=-117.6[/tex]
[tex]\displaystyle S_{xx}=\sum x^2-\frac{(\sum x)^2}{n}[/tex]
[tex]\displaystyle S_{xx}=431-\frac{39}{5}^2=126.8[/tex]
The slope of the linear regression function is given by
[tex]\displaystyle m=\frac{S_{xy}}{S_{xx}}=\frac{-117.6}{126.8}=-0.927[/tex]
The y-intercept ot the linear function is
[tex]\displaystyle b=\bar{y}-b\bar{x}=6.4-(-0.927)(7.8)[/tex]
[tex]\displaystyle b=13.63[/tex]
Thus the best-fit line is
[tex]\displaystyle y=-0.927x+13.63[/tex]
The correct option is the last one
