Which of the following statements are true? 1. The coefficient of rank correlation has the same limits as the Karl Pearson’s coefficient of correlation. 2. The coefficient of correlation is independent of the change of origin but not of scale. 3. The covariance between $$X$$ and $$Y$$ is defined as $$\frac{{\sum {xy} }}{n}$$ where, $$x = \left( {X – \overline X } \right),y = \left( {Y – \overline Y } \right)$$ and $$n = $$ no. of paired observations. 4. $${b_{xy}}$$ is called regression coefficient of $$X$$ variable on $$Y$$ variable. 5. If $${b_{xy}}$$ is 0.4 and $${b_{yx}}$$ is 1.6, coefficient of determination would be 0.8. Select the correct answer:

The correct answer is: B. 1, 3 and 4

1. The coefficient of rank correlation has the same limits as the Karl Pearson’s coefficient of correlation.

This is true. The coefficient of rank correlation, also known as Spearman’s rho, is a measure of the correlation between two variables that is based on the ranks of the data points. It has the same limits as the Pearson correlation coefficient, which is a measure of the linear correlation between two variables. The limits of the Pearson correlation coefficient are -1 and 1.

1. The coefficient of correlation is independent of the change of origin but not of scale.

This is true. The coefficient of correlation is a measure of the linear correlation between two variables. It is independent of the change of origin, which means that it does not change if the variables are shifted up or down. However, it is not independent of the scale, which means that it does change if the variables are multiplied by a constant.

1. The covariance between $X$ and $Y$ is defined as $\frac{{\sum {xy} }}{n}$ where, $x = \left( {X – \overline X } \right),y = \left( {Y – \overline Y } \right)$ and $n = $ no. of paired observations.

This is true. The covariance between two variables $X$ and $Y$ is a measure of how much they vary together. It is defined as $\frac{{\sum {xy} }}{n}$ where, $x = \left( {X – \overline X } \right),y = \left( {Y – \overline Y } \right)$ and $n = $ no. of paired observations.

1. $b_{xy}$ is called regression coefficient of $X$ variable on $Y$ variable.

This is true. The regression coefficient of $X$ variable on $Y$ variable is a measure of how much $X$ changes when $Y$ changes. It is denoted by $b_{xy}$.

1. If $b_{xy}$ is 0.4 and $b_{yx}$ is 1.6, coefficient of determination would be 0.8.

This is false. The coefficient of determination is a measure of how much of the variation in $Y$ can be explained by the variation in $X$. It is denoted by $R^2$. It is calculated as follows:

$$R^2 = \frac{\text{variance of }Y\text{ explained by }X}{\text{total variance of }Y}$$

If $b_{xy}$ is 0.4 and $b_{yx}$ is 1.6, then the coefficient of determination would be 0.64.