Respuesta :
Answer:
the least-squares regression line for the two variables
y = 1.5x -25.5
Step-by-step explanation:
Explanation:-
Let the straight line satisfying the general trend of n dots in a scatter diagram
be y = a + b x ... (1)
we have to determine the constants a and b so that (1) gives for each value of 'x' the best estimate for the average value of y in accordance with the principal of least squares
The normal equations of a and b are
∑y =n a +b∑ x
∑x y = a ∑x +b∑x^2
The line of best fit formula is
[tex]y-y^{-} = r \frac{S.D of y }{S.D of x}(x-x^{-} )[/tex]
Now given data
mean of x is x⁻ = 25
standard deviation of x =3
mean of y is y⁻ = 12
standard deviation of y =6
Given Two variables, x and y, have a correlation of 0.75.
r=0.75
The equation of least squares best fit regression line is
[tex]y-12= 0.75\frac{6 }{3}(x-25 )[/tex]
on simplification we get y-12 = 1.5(x-25)
y = 12 + 1.5x - 1.5X25
y = 1.5x -25.5
The least-squares regression line for the two variables is [tex]y = 1.5x -25.5[/tex]
The given parameters are:
[tex]r = 0.75[/tex] -- the correlation
[tex]\bar x =25[/tex] --- the mean of x
[tex]\sigma x =3[/tex] --- the standard deviation of x
[tex]\bar x =12[/tex] -- the mean of y
[tex]\sigma y =6[/tex] --- the standard deviation of y
The least square regression equation is calculated as:
[tex]y - \bar y = r \times \frac{\sigma_y}{\sigma_x}(x - \bar x)[/tex]
This gives
[tex]y - 12 = 0.75 \times \frac{6}{3}(x - 25)[/tex]
Evaluate the quotients
[tex]y - 12 = 0.75 \times 2(x - 25)[/tex]
[tex]y - 12 = 1.5(x - 25)[/tex]
Open the bracket
[tex]y - 12 = 1.5x - 37.5[/tex]
Add 12 to both sides
[tex]y = 1.5x -25.5[/tex]
Hence, the least-squares regression line for the two variables is [tex]y = 1.5x -25.5[/tex]
Read more about least squares regressions at:
https://brainly.com/question/16975425
