Question Number 192255 by Spillover last updated on 01/Feb/24
$${If}\:\theta\:{be}\:{the}\:{acute}\:{angle}\:{between}\:{two}\:{regression} \\ $$$${line}\:{in}\:{the}\:{case}\:{of}\:{two}\:{variables}\:{x}\:{and}\:{y} \\ $$$${Show}\:{that} \\ $$$$\:\:\mathrm{tan}\:\theta=\frac{\mathrm{1}−{r}}{{r}}.\frac{\sigma_{{x}} \sigma_{{y}} }{\sigma_{{x}} ^{\mathrm{2}} +\sigma_{{y}} ^{\mathrm{2}} }\:\:\: \\ $$$${where}\:\:{r},\sigma_{{x}} ,\sigma_{{y}} \:\:{have}\:{their}\:{usual}\:{meanings}. \\ $$$${Explain}\:{the}\:{significance}\:{where}\:{r}=\mathrm{0}\:\:\:{and}\:{r}=\pm{r} \\ $$$$\mathrm{1}/\mathrm{2}/\mathrm{2024} \\ $$