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,$$Y_{s} = \mu_{Y} + \rho \times \frac {\sigma_{Y}}{\sigma_{X}} \times (X - \mu_{X})$$ ------------------(1)[where $$Y_{s}$$ is the estimated value of Y]Clearly,  here,Slope pf the line = $$\rho \times \frac {\sigma_{Y}}{\sigma_{X}}$$---------------------------------(2)Y- intercept = $$\mu_{Y} - \rho \times \mu_{X} \times \frac {\sigma_{Y}}{\sigma_{X}}$$-----------------(3)and,X - intercept = $$\mu_{X} - \mu_{Y} \times \frac {\sigma_{X}}{\rho \times \sigma_{Y}}$$----------------(4) [putting $$Y_{s} = 0$$ in (1) and taking the value of X]So, since $$\sigma_{Y}, \sigma_{X} > 0$$ [since $$\sigma_{Y} = 0$$ or $$\sigma_{X} = 0$$ will result in a degenerate distribution, hence these cases are discarded]so, correlation coefficient = $$\rho$$ > 0 impliesA. positive slope. [as evident from (1)]clearly from (3) and (4) all the other options are false.