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The two lines of regression are given by 8X +10 Y equals to 25 and 16 X plus 5Y equals to 12, if the variance of X is 25, what is the standard division of Y?
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The two lines of regression are given by 8X +10 Y equals to 25 and 16 ...
Understanding the Lines of Regression
The two lines of regression given are:
1. **8X + 10Y = 25** (Regression of Y on X)
2. **16X + 5Y = 12** (Regression of X on Y)
These equations help in understanding the relationship between variables X and Y.
Calculating the Slopes
To find the slopes of the regression lines:
- For the first line (Y on X):
Rearranging gives:
**Y = (-8/10)X + 2.5**
Thus, slope \(b_{YX} = -0.8\).
- For the second line (X on Y):
Rearranging gives:
**X = (-5/16)Y + 0.75**
Thus, slope \(b_{XY} = -0.3125\).
Using the Formula for Variances
The relationship between the variances and covariance of X and Y can be expressed using the slopes:
\[ b_{YX} = \frac{\text{Cov}(X,Y)}{\text{Var}(X)} \]
\[ b_{XY} = \frac{\text{Cov}(X,Y)}{\text{Var}(Y)} \]
Given:
- \( \text{Var}(X) = 25 \)
- From the slopes:
\( b_{YX} = -0.8 \)
\( b_{XY} = -0.3125 \)
Finding Covariance
Using the first slope:
\[ -0.8 = \frac{\text{Cov}(X,Y)}{25} \]
Thus, \( \text{Cov}(X,Y) = -20 \).
Finding Variance of Y
Using the second slope:
\[ -0.3125 = \frac{-20}{\text{Var}(Y)} \]
Solving gives:
\[ \text{Var}(Y) = 64 \].
Calculating Standard Deviation of Y
Finally, the standard deviation \( \sigma_Y \) is the square root of the variance:
\[ \sigma_Y = \sqrt{64} = 8 \].
Thus, the standard deviation of Y is **8**.
The two lines of regression given are:
1. **8X + 10Y = 25** (Regression of Y on X)
2. **16X + 5Y = 12** (Regression of X on Y)
These equations help in understanding the relationship between variables X and Y.
Calculating the Slopes
To find the slopes of the regression lines:
- For the first line (Y on X):
Rearranging gives:
**Y = (-8/10)X + 2.5**
Thus, slope \(b_{YX} = -0.8\).
- For the second line (X on Y):
Rearranging gives:
**X = (-5/16)Y + 0.75**
Thus, slope \(b_{XY} = -0.3125\).
Using the Formula for Variances
The relationship between the variances and covariance of X and Y can be expressed using the slopes:
\[ b_{YX} = \frac{\text{Cov}(X,Y)}{\text{Var}(X)} \]
\[ b_{XY} = \frac{\text{Cov}(X,Y)}{\text{Var}(Y)} \]
Given:
- \( \text{Var}(X) = 25 \)
- From the slopes:
\( b_{YX} = -0.8 \)
\( b_{XY} = -0.3125 \)
Finding Covariance
Using the first slope:
\[ -0.8 = \frac{\text{Cov}(X,Y)}{25} \]
Thus, \( \text{Cov}(X,Y) = -20 \).
Finding Variance of Y
Using the second slope:
\[ -0.3125 = \frac{-20}{\text{Var}(Y)} \]
Solving gives:
\[ \text{Var}(Y) = 64 \].
Calculating Standard Deviation of Y
Finally, the standard deviation \( \sigma_Y \) is the square root of the variance:
\[ \sigma_Y = \sqrt{64} = 8 \].
Thus, the standard deviation of Y is **8**.
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The two lines of regression are given by 8X +10 Y equals to 25 and 16 X plus 5Y equals to 12, if the variance of X is 25, what is the standard division of Y?
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The two lines of regression are given by 8X +10 Y equals to 25 and 16 X plus 5Y equals to 12, if the variance of X is 25, what is the standard division of Y? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The two lines of regression are given by 8X +10 Y equals to 25 and 16 X plus 5Y equals to 12, if the variance of X is 25, what is the standard division of Y? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The two lines of regression are given by 8X +10 Y equals to 25 and 16 X plus 5Y equals to 12, if the variance of X is 25, what is the standard division of Y?.
The two lines of regression are given by 8X +10 Y equals to 25 and 16 X plus 5Y equals to 12, if the variance of X is 25, what is the standard division of Y? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The two lines of regression are given by 8X +10 Y equals to 25 and 16 X plus 5Y equals to 12, if the variance of X is 25, what is the standard division of Y? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The two lines of regression are given by 8X +10 Y equals to 25 and 16 X plus 5Y equals to 12, if the variance of X is 25, what is the standard division of Y?.
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