Math Problem Statement
Solution
To compute the least-squares regression line for predicting the diastolic pressure () from the systolic pressure (), the steps are as follows:
1. Collect Data
We have the data points for systolic () and diastolic () pressures:
\text{Systolic (x)}: & \; 157, 154, 134, 115, 113 \\ \text{Diastolic (y)}: & \; 103, 94, 87, 83, 77 \\ \text{Systolic (x)}: & \; 123, 119, 118, 130, 116 \\ \text{Diastolic (y)}: & \; 77, 69, 88, 76, 70 \\ \end{aligned}$$ ### 2. Formula for Regression Line The regression line is defined as: $$\hat{y} = a + bx$$ Where: - $$b = \frac{\text{Cov}(x, y)}{\text{Var}(x)}$$ is the slope. - $$a = \bar{y} - b\bar{x}$$ is the intercept. $$\bar{x}$$ and $$\bar{y}$$ are the means of $$x$$ and $$y$$, respectively. ### 3. Compute Necessary Quantities To compute: - $$b = \frac{\sum(x - \bar{x})(y - \bar{y})}{\sum(x - \bar{x})^2}$$ - $$a = \bar{y} - b\bar{x}$$ I will calculate the values numerically for you. Let me handle these calculations. numpy as np # Data points for systolic (x) and diastolic (y) pressures x = np.array([157, 154, 134, 115, 113, 123, 119, 118, 130, 116]) # Systolic y = np.array([103, 94, 87, 83, 77, 77, 69, 88, 76, 70]) # Diastolic # Means of x and y x_mean = np.mean(x) y_mean = np.mean(y) # Slope (b) b = np.sum((x - x_mean) * (y - y_mean)) / np.sum((x - x_mean) ** 2) # Intercept (a) a = y_mean - b * x_mean a, bThe least-squares regression line for predicting the diastolic pressure from the systolic pressure is: $$\hat{y} = 14.1726 + 0.5334x$$ Where: - Slope ($$b$$) = $$0.5334$$ - Intercept ($$a$$) = $$14.1726$$ Would you like further details or assistance with another part? ### Related Questions: 1. How is the slope of a regression line interpreted in this context? 2. How do we compute the correlation coefficient between systolic and diastolic pressures? 3. What is the predicted diastolic pressure for a systolic pressure of 120 mmHg? 4. How would we check if this regression line is a good fit? 5. Can we extend this regression model to nonlinear relationships? ### Tip: Always verify the accuracy of regression predictions by checking residuals (differences between observed and predicted values).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Statistics
Linear Regression
Data Analysis
Formulas
Regression line formula: \hat{y} = a + bx
Slope formula: b = \frac{Cov(x, y)}{Var(x)}
Intercept formula: a = \bar{y} - b\bar{x}
Theorems
Least-Squares Regression Theorem
Suitable Grade Level
Grades 10-12
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