We will use linear regression to find the best-fitting line for the given data points. The equation of a linear regression line is given by:

$p(t) = mt + b$

where:

• $m$ is the slope
• $b$ is the y-intercept

We will also calculate the correlation coefficient $r$ to determine how well the linear model fits the data. Let’s compute these values. numpy as np import matplotlib.pyplot as plt from scipy.stats import linregress

Given data points

t = np.array([0, 2, 4, 6, 8, 10]) p = np.array([0.68, 0.70, 0.90, 1.25, 1.50, 1.90])

Perform linear regression

slope, intercept, r_value, _, _ = linregress(t, p)

Equation of regression line

regression_eq = f"p(t) = {slope:.2f}t + {intercept:.2f}" correlation_coefficient = round(r_value, 2)

Generate regression line points

t_line = np.linspace(min(t), max(t), 100) p_line = slope * t_line + intercept

Plot data points and regression line

plt.scatter(t, p, color='red', label='Data points') plt.plot(t_line, p_line, color='blue', label=f'Regression Line: {regression_eq}') plt.xlabel('Year (t)') plt.ylabel('Price (p) in $ million') plt.title('Linear Regression of Housing Costs') plt.legend() plt.grid(True) plt.show()

regression_eq, correlation_coefficientThe linear regression equation is:

$p(t) = 0.13t + 0.52$

The correlation coefficient is:

$r = 0.97$

This high correlation value suggests a strong positive linear relationship between year $t$ and the apartment price $p$.

Would you like a breakdown of the calculations or interpretation of the results?

Here are five related questions to explore further:

1. How can we interpret the slope of the regression line in terms of housing prices?
2. What does the correlation coefficient tell us about the strength of the relationship?
3. How would we use this model to predict the price of an apartment in 2010?
4. How does linear regression differ from polynomial regression for this data?
5. What are the limitations of using this linear model for future predictions?

Tip: If the correlation coefficient is close to 1 or -1, the linear model is a good fit for the data!