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 implies
A. positive slope. [as evident from (1)]
clearly from (3) and (4) all the other options are false.
