Question

In a partially destroyed laboratory, record of an analysis of correlation of data,
only the following results are legible:
Variance of X = 9
Regression equations are
(i) 8x −10y + 66 = 0
(ii) 40x −18y − 214 = 0
Find out the following missing results.
(i) The means of X and Y
(ii) The coefficient of correlation between x and y
(iii) The standard deviation of Y

Answer 1

Answer on Question #55140 – Math – Statistics and Probability

In a partially destroyed laboratory, record of an analysis of correlation of data, only the following results are legible:

Variance of $X = 9$

Regression equations are

$(^{*})$ $8x - 10y + 66 = 0$

$(^{**})$ $40x - 18y - 214 = 0$

Find out the following missing results.

(i) The means of X and Y

(ii) The coefficient of correlation between x and y

(iii) The standard deviation of Y

Solution

(i) Since two regression lines always intersect at a point $(\overline{x},\overline{y})$ representing mean values of the values $\overline{x}$ and $\overline{y}$ as shown below:

$$\begin{array}{l} 8x - 10y = -66 \\ 40x - 18y = 214 \\ \end{array}$$

Multiplying the first equation by 5 and subtracting from the second, we have

$$32y = 544 \Rightarrow \bar{y} = 17$$

Then $\overline{x} = (10\overline{y} - 66) / 8 = (10 \cdot 17 - 66) / 8 = 13$

(ii) To find the given regression equations in such a way that the coefficient of dependent variable is less than one at least in one equation.

So, $8x - 10y = -66 \Rightarrow 10y = 66 + 8x \Rightarrow y = \frac{66}{10} + \frac{8}{10} x$ .

That is, $b_{yx} = 8 / 10 = 0.8$

And $40x - 18y = 214 \Rightarrow 40x = 214 + 18y \Rightarrow x = \frac{214}{40} + \frac{18}{40} y$

That is, $b_{yx} = 18 / 40 = 0.45$ .

Hence coefficient of correlation $r$ between $x$ and $y$ is given by:

$$r = \sqrt {b _ {x y} \times b _ {y x}} = \sqrt {0 . 4 5 \cdot 0 . 8 0} = 0. 6 0$$

(iii) To determine the standard deviation of $y$ , consider the formula:

$$\sigma_ {y} = \frac {b _ {y x} \sigma_ {x}}{r} = \frac {0 . 8 \cdot \sqrt {9}}{0 . 6} = 4.$$

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