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For an experiment we have the following data set: n = 4, ∑x = a, ∑y = 10, ∑xy = 21, ∑x2 = 30, ∑y2 = 30. If the correlation coefficient is -0.8 then the value of a is:
- a)7
- b)10
- c)8
- d)9
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
For an experiment we have the following data set: n = 4,∑x = a,&su...
Explanation:
Given data:
- n = 4
- ∑x = a
- ∑y = 10
- ∑xy = 21
- ∑x^2 = 30
- ∑y^2 = 30
- Correlation coefficient (r) = -0.8
Formula for correlation coefficient (r):
r = (n∑xy - ∑x∑y) / √[(n∑x^2 - (∑x)^2)(n∑y^2 - (∑y)^2)]
Substitute the given values:
- r = -0.8
- n = 4
- ∑x = a
- ∑y = 10
- ∑xy = 21
- ∑x^2 = 30
- ∑y^2 = 30
Calculate the correlation coefficient:
- -0.8 = (4*21 - a*10) / √[(4*30 - a^2)(4*30 - 100)]
- -0.8 = (84 - 10a) / √[(120 - a^2)(120 - 100)]
- -0.8 = (84 - 10a) / √[(120 - a^2)(20)]
Solve for 'a':
- -0.8 = (84 - 10a) / √[2400 - 20a + a^2]
- -0.8 = (84 - 10a) / √[2400 - 20a + a^2]
- -0.8 = (84 - 10a) / √(a^2 - 20a + 2400)
- -0.8*√(a^2 - 20a + 2400) = 84 - 10a
- -0.8(a^2 - 20a + 2400) = (84 - 10a)^2
- -0.8a^2 + 16a - 1920 = 100 - 168a + 100a^2
- 100a^2 - 0.8a^2 + 100a - 168a - 1920 - 100 = 0
- 99.2a^2 - 68a - 2020 = 0
- 99.2a^2 - 198.8a + 130.8a - 2020 = 0
- 99.2a(a - 2) + 130.8(a - 2) = 0
- (99.2a + 130.8)(a - 2) = 0
Therefore, a = 10 or a = -130.8/99.2 = -1.313
Conclusion:
Since the value of 'a' cannot be negative, the correct value of 'a' is 10.
Given data:
- n = 4
- ∑x = a
- ∑y = 10
- ∑xy = 21
- ∑x^2 = 30
- ∑y^2 = 30
- Correlation coefficient (r) = -0.8
Formula for correlation coefficient (r):
r = (n∑xy - ∑x∑y) / √[(n∑x^2 - (∑x)^2)(n∑y^2 - (∑y)^2)]
Substitute the given values:
- r = -0.8
- n = 4
- ∑x = a
- ∑y = 10
- ∑xy = 21
- ∑x^2 = 30
- ∑y^2 = 30
Calculate the correlation coefficient:
- -0.8 = (4*21 - a*10) / √[(4*30 - a^2)(4*30 - 100)]
- -0.8 = (84 - 10a) / √[(120 - a^2)(120 - 100)]
- -0.8 = (84 - 10a) / √[(120 - a^2)(20)]
Solve for 'a':
- -0.8 = (84 - 10a) / √[2400 - 20a + a^2]
- -0.8 = (84 - 10a) / √[2400 - 20a + a^2]
- -0.8 = (84 - 10a) / √(a^2 - 20a + 2400)
- -0.8*√(a^2 - 20a + 2400) = 84 - 10a
- -0.8(a^2 - 20a + 2400) = (84 - 10a)^2
- -0.8a^2 + 16a - 1920 = 100 - 168a + 100a^2
- 100a^2 - 0.8a^2 + 100a - 168a - 1920 - 100 = 0
- 99.2a^2 - 68a - 2020 = 0
- 99.2a^2 - 198.8a + 130.8a - 2020 = 0
- 99.2a(a - 2) + 130.8(a - 2) = 0
- (99.2a + 130.8)(a - 2) = 0
Therefore, a = 10 or a = -130.8/99.2 = -1.313
Conclusion:
Since the value of 'a' cannot be negative, the correct value of 'a' is 10.
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