To solve this problem, we will:

1. Calculate the regression equations: $Y$ on $X$ and $X$ on $Y$.
2. Calculate the coefficient of correlation, $r$.

Steps:

1. Regression Equations:

The equations are:

• $Y = a + bX$ (Regression of $Y$ on $X$).
• $X = c + dY$ (Regression of $X$ on $Y$).

We need:

• Means: $\bar{X}$ and $\bar{Y}$,
• $S_{XY}$: Covariance,
• $S_{XX}$: Variance of $X$,
• $S_{YY}$: Variance of $Y$.

2. Coefficient of Correlation:

The formula for $r$ is: $r = \frac{S_{XY}}{\sqrt{S_{XX} \cdot S_{YY}}}$

Let me calculate these values.import numpy as np import pandas as pd

Data

X = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9]) Y = np.array([9, 8, 10, 12, 11, 13, 14, 16, 15])

Calculations

n = len(X) mean_X = np.mean(X) mean_Y = np.mean(Y)

Covariance and Variances

S_XY = np.sum((X - mean_X) * (Y - mean_Y)) S_XX = np.sum((X - mean_X) ** 2) S_YY = np.sum((Y - mean_Y) ** 2)

Regression coefficients

b = S_XY / S_XX # Slope of Y on X a = mean_Y - b * mean_X # Intercept of Y on X

d = S_XY / S_YY # Slope of X on Y c = mean_X - d * mean_Y # Intercept of X on Y

Coefficient of correlation

r = S_XY / np.sqrt(S_XX * S_YY)

Regression equations

regression_Y_on_X = f"Y = {a:.2f} + {b:.2f}X" regression_X_on_Y = f"X = {c:.2f} + {d:.2f}Y"

regression_Y_on_X, regression_X_on_Y, r('Y = 7.25 + 0.95X', 'X = -6.40 + 0.95Y', 0.95)('Y = 7.25 + 0.95X', 'X = -6.40 + 0.95Y', 0.95)success### Results:

1. Regression Equation of $Y$ on $X$: $Y = 7.25 + 0.95X$

2. Regression Equation of $X$ on $Y$: $X = -6.40 + 0.95Y$

3. Coefficient of Correlation ($r$): $r = 0.95$

The correlation coefficient indicates a strong positive linear relationship between $X$ and $Y$.

Would you like a detailed step-by-step explanation or further analysis?

Related Questions:

1. What does the correlation coefficient signify in this context?
2. How can you use the regression equations for predictions?
3. What are the assumptions behind linear regression?
4. How would the results change if an outlier were introduced?
5. Can regression equations be used for non-linear relationships?

Tip:

Always visualize your data with a scatter plot to verify if a linear regression model is appropriate.