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.
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Important Questions from Correlation Analysis
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:
X, Y and Z are three uncorrelated variables having variances \(\sigma_x^2, \sigma_y^2 \:and\:\sigma_z^2\) respectively, then the correlation between X + Y and Y + Z is:
If X ∼ N (0, 1) and Y = X2 then the correlation coefficient r (X, Y) is
Calculate the correlation coefficient between the following values :
x: 3, 5, 1, 7, 5
y: 4, 3, 0, 8, 2
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).
