Model Answer
0 min readIntroduction
Regression analysis is a powerful statistical tool used to model the relationship between a dependent variable and one or more independent variables. The regression lines represent the best-fit lines that describe this relationship. Understanding the properties of these lines, such as their slopes and intercepts, allows us to calculate important statistical measures like variance and covariance, which quantify the spread and association of the variables. This question requires applying these concepts to determine the variance of X and the covariance between X and Y, given the regression equations and the variance of Y.
Understanding the Regression Lines
We are given two regression lines:
- Y = X + 5 (Regression of Y on X)
- 16X = 9Y – 94 (Regression of X on Y)
These can be rewritten as:
- Y = 1X + 5 (byx = 1)
- X = (9/16)Y – (94/16) = (9/16)Y – 5.875 (bxy = 9/16)
Here, byx represents the regression coefficient of Y on X, and bxy represents the regression coefficient of X on Y.
Calculating the Correlation Coefficient (r)
The correlation coefficient (r) is related to the regression coefficients by the following formula:
r2 = bxy * byx
Substituting the values, we get:
r2 = (9/16) * 1 = 9/16
Therefore, r = ±√(9/16) = ±3/4 = ±0.75
Finding the Means of X and Y
The regression line Y = X + 5 passes through the point (X̄, Ȳ), where X̄ and Ȳ are the means of X and Y respectively. Therefore:
Ȳ = X̄ + 5
Similarly, the regression line X = (9/16)Y – 5.875 passes through the point (X̄, Ȳ). Therefore:
X̄ = (9/16)Ȳ – 5.875
Substituting Ȳ = X̄ + 5 into the second equation:
X̄ = (9/16)(X̄ + 5) – 5.875
X̄ = (9/16)X̄ + (45/16) – 5.875
X̄ = (9/16)X̄ + 2.8125 – 5.875
X̄ = (9/16)X̄ – 3.0625
X̄ – (9/16)X̄ = -3.0625
(7/16)X̄ = -3.0625
X̄ = (-3.0625 * 16) / 7 = -7.00
Now, we can find Ȳ:
Ȳ = X̄ + 5 = -7 + 5 = -2
Calculating the Variance of X
We know that the variance of Y (Var(Y)) is 16. The relationship between the variances and the correlation coefficient is given by:
Var(X) = r2 * Var(Y)
Using r = ±0.75, we have r2 = 0.5625
Var(X) = 0.5625 * 16 = 9
Calculating the Covariance between X and Y
The covariance between X and Y (Cov(X, Y)) is given by:
Cov(X, Y) = r * σX * σY
Where σX is the standard deviation of X and σY is the standard deviation of Y.
σX = √Var(X) = √9 = 3
σY = √Var(Y) = √16 = 4
Cov(X, Y) = ±0.75 * 3 * 4 = ±9
Conclusion
In conclusion, we have determined the variance of X to be 9 and the covariance between X and Y to be ±9. The sign of the covariance depends on the sign of the correlation coefficient, indicating whether the variables tend to increase or decrease together. This analysis demonstrates the application of regression principles to derive key statistical measures from the given regression lines and variance information.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.