MHB Least squares regression line (I'm very lost)

[Melody55]

Hi! Basically this is the exercise:

Given the covariance of x and y is -12 and the variance of x is 6,5, using the least squares line of best fit connecting x and y yo estimate the value of x when y=15

x	2	5	9	7	9	10	7
y	25	17	11	10	8	7	13

any help would mean everything, I'm desperate :(

 

[HOI]

Do you know what "least squares best fit" means?
It is the line y= ax+ b that "best fits" in very specific way. When x= 2, that equation gives y= 2a+ b while the correct value is 25. The "error", if any, is 2a+ b- 25. If we want to find a "total error" by adding those, some might be negative and cancel positive errors giving too small a total error. We could fix that by taking the absolute value but the absolute value function is not differentiable at 0. So instead we fix the sign problem by squaring. The "square error" at x= 2 is $(2a+ b- 25)^2$.

Using all of the given data,

$(2a+ b- 25)^2$

$(5a+ b- 17)^2$

$(9a+ b- 11)^2$

$(7a+ b- 10)^2$

$(9a+ b- 8)^2$

$(10a+ b- 7)^2$

$(7a+ b- 13)^2$
The total square error is

$(2a+ b- 25)^2+(5a+ b- 17)^2+ (9a+ b- 11)^2+ (7a+ b- 10)^2+ (9a+ b- 8)^2+ (10a+ b- 7)^2+ (7a+ b- 13)^2$.
That's a function of the two variables, a and b. Find the minimum by taking the partial derivatives with respect to a and b and setting them equal to 0,