To compute the least-squares regression line equation for predicting diastolic pressure ($y$) from systolic pressure ($x$), we will need to calculate the slope ($b_1$) and intercept ($b_0$) using the formulas:

1. Slope ($b_1$): $b_1 = \frac{\text{Cov}(x, y)}{\text{Var}(x)} = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sum{(x_i - \bar{x})^2}}$

2. Intercept ($b_0$): $b_0 = \bar{y} - b_1\bar{x}$

3. The equation of the regression line will then be: $y = b_0 + b_1x$

Extracting Data:

From the table provided:

• $x$ (Systolic): $126, 104, 103, 110, 122, 117, 130, 118$
• $y$ (Diastolic): $88, 86, 80, 80, 71, 88, 75, 69$

I will compute these values step by step. numpy as np

Data: Systolic (x) and Diastolic (y)

x = np.array([126, 104, 103, 110, 122, 117, 130, 118]) # Systolic pressure y = np.array([88, 86, 80, 80, 71, 88, 75, 69]) # Diastolic pressure

Calculate means

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

Calculate slope (b1)

numerator = np.sum((x - x_mean) * (y - y_mean)) denominator = np.sum((x - x_mean) ** 2) b1 = numerator / denominator

Calculate intercept (b0)

b0 = y_mean - b1 * x_mean

b0, b1The equation of the least-squares regression line for predicting diastolic pressure ($y$) from systolic pressure ($x$) is:

$y = 101.5438 - 0.1885x$

Details:

• Intercept ($b_0$): 101.5438
• Slope ($b_1$): $-0.1885$

Would you like the detailed calculations for each step? Let me know if you have any questions!

Here are 5 questions to expand:

1. How do we interpret the slope ($-0.1885$) in this context?
2. What would be the predicted diastolic pressure for a systolic pressure of $120$?
3. How can we calculate the coefficient of determination ($R^2$) for this regression line?
4. What assumptions are made when using linear regression for prediction?
5. What are possible limitations of using this linear model for blood pressure prediction?

Tip: Always plot your data to visually check the fit of the regression line!