Question

The regression equation of y on x and that of x on y are
8x −10y + 66 = 0
and 40x −18y = 214 respectively, and the variance of x is 9.
(i) Find σy .

Answer 1

Answer on Question #66775 – Math – Statistics and Probability

Question

The regression equation of $y$ on $x$ and that of $x$ on $y$ are

$8x - 10y + 66 = 0$ and $40x - 18y = 214$ respectively, and the variance of $x$ is 9.

(i) Find $\sigma y$.

Solution

$$\begin{array}{l} 8x - 10y + 66 = 0 \rightarrow y = \frac{4}{5}x + \frac{33}{5} \rightarrow b_{yx} = \frac{4}{5} = 0.8. \\ 40x - 18y = 214 \rightarrow x = \frac{9}{20}y + \frac{107}{20} \rightarrow b_{xy} = \frac{9}{20} = 0.45. \\ b_{yx} = r \frac{\sigma_y}{\sigma_x}, \quad b_{xy} = r \frac{\sigma_x}{\sigma_y} \quad \text{so} \quad b_{yx} b_{xy} = r^2 \rightarrow \\ \rightarrow r = \sqrt{b_{yx} b_{xy}} = \sqrt{0.8 * 0.45} = 0.6. \\ \end{array}$$

Thus, $\sigma_y = r b_{yx} \sigma_x = 0.6 * 0.8 * \sqrt{9} = 1.44$.

Answer: 1.44.

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