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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