Regression coefficients have the same sign as each other (and as $r$); one negative $\Rightarrow$ the other negative.

The correlation coefficient relates to the two regression coefficients by

$r=\pm\sqrt{b_{yx}\cdot b_{xy}}.$

Since $r^2=b_{yx}\cdot b_{xy}$ and $r^2\ge 0$, the product $b_{yx}\cdot b_{xy}$ must be $\ge 0$. Two numbers whose product is non-negative must have the same sign. Hence both regression coefficients are either positive together or negative together — they cannot differ in sign.

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Doubling both variables scales $S_x$ and $S_y$ equally, so the ratio $\dfrac{S_y}{S_x}$ and the regression coefficient are unchanged.

GSEB Class-12 Statistics, Linear Regression:

The regression coefficient of $Y$ on $X$ is:

$b_{yx} = r \cdot \frac{S_y}{S_x}$

Let the new variables be $x' = 2x$ and $y' = 2y$. Change of scale does not affect the correlation coefficient $r$, and the standard deviations scale by the same factor:

$S_{x'} = 2 S_x, \qquad S_{y'} = 2 S_y$

Therefore the new regression coefficient is: …

The correlation coefficient $r$ is the geometric mean of the regression coefficients: $r=\pm\sqrt{b_{xy}\times b_{yx}}$.

The two regression coefficients are $b_{yx}=r\dfrac{\sigma_y}{\sigma_x}$ and $b_{xy}=r\dfrac{\sigma_x}{\sigma_y}$. Multiplying them:

$b_{yx}\times b_{xy}=r\frac{\sigma_y}{\sigma_x}\cdot r\frac{\sigma_x}{\sigma_y}=r^2.$

Hence …

A regression coefficient $b>0$ indicates a positive/direct relationship — the two variables move in the same direction.

A regression coefficient measures the average change in the dependent variable for a unit change in the independent variable. When $b>0$, an increase in the independent variable is accompanied by an increase in the dependent variable (and a decrease by a decrease). Thus $b>0$ interprets a positive (direct) relationship be …

The correct statement is that the coefficient of correlation always lies between $-1$ and $+1$; the other options are incorrect or garbled.

Examine each option:

• (a) Garbled/incorrect (there are exactly two lines of regression, not "at the most more").
• (b) Correct — Karl Pearson's coefficient of correlation $r$ always satisfies $-1 \le r \le +1$.
• (c) Incorrect — in a scatter diagram a downward trend indicates negative correlation, not positive. …