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The correlation coefficient between two variables Xand Yis found to be 0.6. All the observations on X and Y are transformed using the transformations U= 2 - 3X and V= 4 Y + 1. The correlation coefficient between the transformed variables U and Kwill be
- a)-0.5
- b)+0.5
- c)-0.6
- d)+0.6
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The correlation coefficient between two variables Xand Yis found to be...
Correlation Coefficient:
The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where -1 indicates a perfect negative relationship, +1 indicates a perfect positive relationship, and 0 indicates no relationship.
Given Information:
The correlation coefficient between variables X and Y is 0.6.
Transformation:
The observations on X and Y are transformed using the transformations U=2-3X and V=4Y-1.
Calculating the Correlation Coefficient:
To find the correlation coefficient between the transformed variables U and V, we need to calculate the covariance and standard deviations of U and V.
Covariance Calculation:
The covariance between U and V can be calculated using the formula:
Cov(U,V) = E[(U - E[U])(V - E[V])]
Since E[U] = E[V] = 0 (due to the transformation), the formula simplifies to:
Cov(U,V) = E[UV]
Calculating E[UV]:
E[UV] = E[(2-3X)(4Y-1)]
= E[8Y - 2 - 12XY + 3X]
The expected value of a constant is the constant itself, so we can simplify further:
E[UV] = 8E[Y] - 2 - 12E[XY] + 3E[X]
Calculating E[X], E[Y], E[XY]:
To calculate E[X], E[Y], and E[XY], we need the original data. However, the question does not provide the values of X and Y.
Conclusion:
Since we do not have the original data, we cannot calculate the expected values E[X], E[Y], and E[XY]. Therefore, we cannot determine the covariance Cov(U,V) and the correlation coefficient between the transformed variables U and V.
Hence, the correct answer cannot be determined from the given information.
The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where -1 indicates a perfect negative relationship, +1 indicates a perfect positive relationship, and 0 indicates no relationship.
Given Information:
The correlation coefficient between variables X and Y is 0.6.
Transformation:
The observations on X and Y are transformed using the transformations U=2-3X and V=4Y-1.
Calculating the Correlation Coefficient:
To find the correlation coefficient between the transformed variables U and V, we need to calculate the covariance and standard deviations of U and V.
Covariance Calculation:
The covariance between U and V can be calculated using the formula:
Cov(U,V) = E[(U - E[U])(V - E[V])]
Since E[U] = E[V] = 0 (due to the transformation), the formula simplifies to:
Cov(U,V) = E[UV]
Calculating E[UV]:
E[UV] = E[(2-3X)(4Y-1)]
= E[8Y - 2 - 12XY + 3X]
The expected value of a constant is the constant itself, so we can simplify further:
E[UV] = 8E[Y] - 2 - 12E[XY] + 3E[X]
Calculating E[X], E[Y], E[XY]:
To calculate E[X], E[Y], and E[XY], we need the original data. However, the question does not provide the values of X and Y.
Conclusion:
Since we do not have the original data, we cannot calculate the expected values E[X], E[Y], and E[XY]. Therefore, we cannot determine the covariance Cov(U,V) and the correlation coefficient between the transformed variables U and V.
Hence, the correct answer cannot be determined from the given information.
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The correlation coefficient between two variables Xand Yis found to be 0.6. All the observations on X and Y are transformed using the transformations U= 2 - 3X and V= 4 Y + 1. The correlation coefficient between the transformed variables U and Kwill bea)-0.5b)+0.5c)-0.6d)+0.6Correct answer is option 'A'. Can you explain this answer?
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