To find the regression lines for the given data, we calculate the equations of the best-fit line for the relationship between $x$ and $y$. These equations are usually in the form:

1. Regression of $y$ on $x$:
   $y = a + bx$

2. Regression of $x$ on $y$:
   $x = c + dy$

Where:

• $b = \frac{\text{Cov}(x, y)}{\text{Var}(x)}$
• $d = \frac{\text{Cov}(x, y)}{\text{Var}(y)}$
• $a = \bar{y} - b\bar{x}$
• $c = \bar{x} - d\bar{y}$

Let me calculate these values for your data. numpy as np

Data points

x = np.array([10, 14, 19, 26, 30, 34, 39]) y = np.array([12, 16, 18, 26, 29, 35, 38])

Means of x and y

x_mean = np.mean(x) y_mean = np.mean(y)

Variance and covariance

var_x = np.var(x, ddof=0) var_y = np.var(y, ddof=0) cov_xy = np.cov(x, y, ddof=0)[0, 1]

Regression coefficients

b = cov_xy / var_x # Slope of y on x a = y_mean - b * x_mean # Intercept of y on x

d = cov_xy / var_y # Slope of x on y c = x_mean - d * y_mean # Intercept of x on y

Regression equations

reg_y_on_x = (a, b) # y = a + bx reg_x_on_y = (c, d) # x = c + dy

reg_y_on_x, reg_x_on_yThe regression equations for the given data are:

1. Regression of $y$ on $x$:
   $y = 2.27 + 0.92x$

2. Regression of $x$ on $y$:
   $x = -2.14 + 1.07y$

Let me know if you would like further details about these calculations!

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Related Questions:

1. What is the significance of the slope in each regression equation?
2. How can we interpret the intercept values in the regression equations?
3. What is the correlation coefficient for this dataset?
4. Can we use these equations to predict $y$ for any $x$, and vice versa?
5. How would outliers in the data affect the regression lines?

Tip:

To ensure accurate predictions, always verify that your dataset follows the assumptions of linear regression, such as linearity and homoscedasticity.