MATH SOLVE

2 months ago

Q:
# The least squares line of best fit for a data set with a positive correlation coefficient always has a: A. positive slope. B. positive x-intercept. C. positive y-intercept. D. Both A and C are correct.

Accepted Solution

A:

Answer:A. positive slope.Step-by-step explanation:In the least square linear regression of Y on X, the straight line of best fit is given by,[tex]Y_{s} = \mu_{Y} + \rho \times \frac {\sigma_{Y}}{\sigma_{X}} \times (X - \mu_{X})[/tex] ------------------(1)[where [tex]Y_{s}[/tex] is the estimated value of Y]Clearly, here,Slope pf the line = [tex]\rho \times \frac {\sigma_{Y}}{\sigma_{X}}[/tex]---------------------------------(2)Y- intercept = [tex]\mu_{Y} - \rho \times \mu_{X} \times \frac {\sigma_{Y}}{\sigma_{X}}[/tex]-----------------(3)and,X - intercept = [tex]\mu_{X} - \mu_{Y} \times \frac {\sigma_{X}}{\rho \times \sigma_{Y}}[/tex]----------------(4) [putting [tex]Y_{s} = 0[/tex] in (1) and taking the value of X]So, since [tex]\sigma_{Y}, \sigma_{X} > 0[/tex] [since [tex]\sigma_{Y} = 0[/tex] or [tex]\sigma_{X} = 0[/tex] will result in a degenerate distribution, hence these cases are discarded]so, correlation coefficient = [tex]\rho[/tex] > 0 impliesA. positive slope. [as evident from (1)]clearly from (3) and (4) all the other options are false.