Calculating $\hat{\beta_1}$ given $\overline{X}$, $\overline{Y}$, their standard deviations and the correlation coefficient.

Question

In a linear regression problem I'm asked to find $\hat{\beta_1}$ given only the following information:

• $\overline{X} = 163.5$
• $\overline{Y} = 874.1$
• $\sigma_X = 16.2$
• $\sigma_Y = 54.2$
• $r = -0.774$

I'm aware that $\beta_1 = \frac{S_{XY}}{S_{XY}}$ but I'm not sure how I would find it given these parameters, maybe there's some formula for $S_{XX}$ (or $S_{XY}$) that involves the $\sigma$s that I'm not awar of?

Answer 1

With $S_{XY}=\sum (X_i-\overline X)(Y_i-\overline Y)$ and $S_{XX}=\sum (X_i-\overline X)^2$, I guess

$$\hat\beta_1=\frac{S_{XY}}{S_{XX}}=r\sqrt{\frac{S_{YY}}{S_{XX}}}$$

Since $$S_{XX}/n=\sigma_X^2\quad,\quad S_{YY}/n=\sigma_Y^2\,,$$

you have $$\frac{S_{YY}}{S_{XX}}=\frac{\sigma^2_Y}{\sigma^2_X}$$

Not suprising, since it is a standard relation that the regression coefficient of $Y$ on $X$ is $$\beta_1=r\frac{\sigma_Y}{\sigma_X}$$