Mathematics: 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.

The least squares line of best fit for a data set

Ответ:

kayranicole1

10.01.2022

Mathematics

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 implies

A. positive slope. [as evident from (1)]

clearly from (3) and (4) all the other options are false.

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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.

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