Question

From the given data:

Variable-	x	y
Mean	6	8
Standard Deviation	4	6
Correlation coefficient	$\dfrac{2}{3}$

Find regression coefficients ${{b}_{yx}}$ and ${{b}_{xy}}$ .

Answer 1

For solving this problem we need to have a clear understanding of what does correlation coefficient, regression coefficients and standard deviation signify. By calculating the regression coefficients using the standard formula, we get the values of ${{b}_{yx}}$ and ${{b}_{xy}}$ respectively.

Complete step-by-step solution:
Unlike range and quartiles, the variance combines all the values in a data set to produce a measure of spread. The standard deviation (the square root of the variance) are the most commonly used measures of spread. We know that variance is a measure of how spread out a data set is. It is calculated as the average squared deviation of each number from the mean of a data set. Standard deviation is the measure of spread most commonly used in statistical practice when the mean is used to calculate central tendency. Thus, it measures spread around the mean. Because of its close links with the mean, standard deviation can be greatly affected if the mean gives a poor measure of central tendency. The correlation coefficient is a statistical measure of the strength of the relationship between the relative movements of two variables. The values range between $-1.0$ and $1.0$ . Regression coefficients are estimates of the unknown population parameters and describe the relationship between a predictor variable and the response.
According to the given problem, we have the mean of x as $6$ , mean of y as $8$ , standard deviations as $4\text{ }\left( {{S}_{x}} \right)$ and $6\text{ }\left( {{S}_{y}} \right)$ respectively for x and y. The correlation coefficient (r) is given as $\dfrac{2}{3}$ . Using the standard formula for regression coefficients, we get that,
${{b}_{yx}}=\dfrac{r}{S_{x}^{2}}=\dfrac{\dfrac{2}{3}}{{{4}^{2}}}=\dfrac{1}{24}$
${{b}_{xy}}=\dfrac{r}{S_{y}^{2}}=\dfrac{\dfrac{2}{3}}{{{6}^{2}}}=\dfrac{1}{54}$
Thus, the regression coefficients, ${{b}_{yx}}$ and ${{b}_{xy}}$ are $\dfrac{1}{24}$ and $\dfrac{1}{54}$ respectively.

Note: These types of problems may seem simple but there are high chances of miscalculations due to the fact that the values of mean are not required to solve this problem. Hence one may get confused as to which formula must be used. We need to carefully perform the calculations after using the correct formula.