For an experiment we have the following data set: n = 4, ∑x = a, ∑y = 10, ∑xy = 21, ∑x 2 = 30, ∑y 2 = 30. If the correlation coefficient is -0.8 then the value of a is:
10
Finding 'a' from the Correlation Coefficient
We are given n = 4, \(\sum x = a\), \(\sum y = 10\), \(\sum xy = 21\), \(\sum x^2 = 30\), \(\sum y^2 = 30\), and r = -0.8. We must find a.
Pearson Correlation Coefficient Formula
The Pearson correlation coefficient is given by:
\(r = \dfrac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}\)
Substituting the Values
Plugging in the given data:
\(-0.8 = \dfrac{4(21) - (a)(10)}{\sqrt{[4(30) - a^2][4(30) - 10^2]}}\)
\(-0.8 = \dfrac{84 - 10a}{\sqrt{(120 - a^2)(20)}}\)
Testing the Options
Rather than solving the resulting equation directly, we substitute each option for a.
Option: a = 10
\(r = \dfrac{84 - 10(10)}{\sqrt{20(120 - 100)}} = \dfrac{-16}{\sqrt{20 \times 20}} = \dfrac{-16}{20} = -0.8\)
This matches the given correlation coefficient exactly.
Option: a = 7
\(r = \dfrac{84 - 70}{\sqrt{20(120 - 49)}} = \dfrac{14}{\sqrt{1420}} \approx 0.37\) — does not match.
Option: a = 8
\(r = \dfrac{84 - 80}{\sqrt{20(120 - 64)}} = \dfrac{4}{\sqrt{1120}} \approx 0.12\) — does not match.
Option: a = 9
\(r = \dfrac{84 - 90}{\sqrt{20(120 - 81)}} = \dfrac{-6}{\sqrt{780}} \approx -0.21\) — does not match.
Conclusion
Only a = 10 produces the given correlation coefficient r = -0.8. Hence, the value of a is 10.
Similar Questions
The coefficients of the regression β X|y and β Y|x , are known, The coefficient of correlation equals:
Which option is correct for the correlation ratio E 2?
Let θ be the angle made by the line of regression of Y on X. If σ Y= 2σ X and the correlation coefficient between X and Y is 0.3, the value θ equals
The multiple correlation coefficient R 1,23 as compared to any simple correlation coefficients between the distinct variable X 1 ,X 2, and X 3is
The correlation coefficient between two variables X and Y is 0.4. The correlation coefficient between 2X and (-Y) will be:
The coefficient of correlation between two variables X and Y is 0.48. The covariance is 36. The variance of X is 16. The standard deviation of Y is:
If the regression line of Y on X is Y = 30 - 0.9X and the standard deviations are S x= 2 and S y= 9, then the value of the correlation coefficient r xy is :
If a data set contains n paired values on two variables x(independent) and y(dependent), then their plot is called:
Suppose r xy is the correlation coefficient between two variables X and Y
where s.d.(X) = s.d.(Y). If θ is the angle between the two regression lines of Y on X and X on Y then:The limit of multiple correlation coefficient R 1.23 are:
Important Questions from Correlation Analysis
If X ∼ N (0, 1) and Y = X2 then the correlation coefficient r (X, Y) is
Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): If the securities with less than perfect negative correlation between their price movements are combined, portfolio risk can be reduced significantly.
Reason (R): The term with negative correlation has the effect of reducing the computed value of total portfolio risk, given other terms that are positive.
In the light of the above statements, choose the most appropriate answer from the options given below:
Consider two exponentially distributed random variables X and Y, both having a mean of 0.50. Let Z = X + Y and r be the correlation coefficient between X and Y. If the variance of Z equals 0, then the value of r is _______ (round off to 2 decimal places).
The two-regression equation of variable \(\rm{x}\) and \(\rm{y}\) are
\(\rm{y = 0.8x + 9.8}\) and \(\rm{x = 10.2 + 0.6y}\)
The coefficient of correlation between \(\rm{x}\) and \(\rm{y}\) is
- In which conditions, Karl Pearson's correlation coefficient can be calculated ?
A. If means of both the variables are equal
B. If one variable is measured in interval scale and another is measured in ordinal scale
C. If there is linear relationship between two variables
D. If data are obtained in interval or ratio scale for both the variables
E. If direction of relationship between two variables is known
Choose the most appropriate answer from the options given below :
