Correlation Coefficient| Notes|Example|Hints|Solution

Today we are providing notes on the Correlation Coefficient.

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Problem Set: 

1. Calculate the coefficient of correlation between X and Y for the following:

X	Y
1	2
3	6
4	8
5	10
7	14
8	16
10	20

2. In the two sets of variables X and Y with 50 observations each, the following data were observed:


$\small \bar X=10,\sigma_{x}= 3,\sigma_{y}=2,r(X,Y)=0.3$, but on the subsequent verification it was found that one value of X(=10) and the corresponding value of Y(=6) were inaccurate and hence weeded out. With the remaining 49 pairs of values, how is the original value of r affected?


3. $\small X_{1}$ and $\small X_{2}$ are independent variables with means 5 and 10 and standard deviations 2 and 3 respectively. Obtain $\small r(U,V)$ where $\small U=3X_{1}+4 X_{2}$ and $\small V= 3X_{1}-X_{2}$.


4. If $\small X$ and $\small Y$ and normal and independent with zero means and standard deviations 9 and 12 respectively, and If $\small X+2Y$ and $\small kX-Y$ are uncorrelated, then find k.


5. If $\small V(X)= V(Y)= \sigma^{2}$, $\small Cov(X,Y)=\frac{\sigma^{2}}{2}$ find
(i) $\small Var(2X-3Y)$
(ii) $\small Corr(2X+3,2Y-3)$


6. Two independent variable X and Y have the following variances $\small \sigma_{x}^2=36,\sigma_{y}^{2}=16$. Calculate the coefficient of correlation between $\small U=X+Y$ and $\small V=X-Y$

7. If $\small X,Y$ are standardised random variables, $\small E(X)=E(Y)=0$ and $\small V(X)=V(Y)=1$ then prove that $\small r(aX+bY,bX+aY)=\frac{1+2ab}{a^2+b^2}$.

8. The variables X and Y are connected by the equation $\small aX+bY+c=0$. Show that the correlation between them is -1, if the signs of a and b are alike and +1 if they are different. 

• Spearman's Rank Correlation Coefficient

Tags: rank correlation, spearman's rank correlation, rank correlation example, rank correlation solution, correlation coefficient, covariance, variance, range of correlation efficient, correlation coefficient problems, correlation coefficient solution set, correlation coefficient notes

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