Math Problem Statement
Solution
To solve this problem, we need to calculate the weighted least-squares regression line . The weights will account for the different reliabilities of the data points.
Steps:
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Weight Assignment:
- First and last points: weight = 0.5.
- Interior points: weight = 1.
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Weighted Least-Squares Formulas:
- Let the weights be , the x-values be , and the y-values be .
- The formulas for (slope) and (intercept) are: where and .
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Data: Points and weights:
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Calculate Weighted Means:
- and will be calculated using the weights.
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Apply the Weighted Least-Squares Formulas: Plug in the data, calculate and .
Let me calculate this for you. numpy as np
Data points
x = np.array([-4, -1, 0, 1, 4]) y = np.array([0, 0, 3, 5, 5]) weights = np.array([0.5, 1, 1, 1, 0.5])
Weighted means
weighted_mean_x = np.sum(weights * x) / np.sum(weights) weighted_mean_y = np.sum(weights * y) / np.sum(weights)
Weighted least-squares slope (beta_1)
numerator = np.sum(weights * (x - weighted_mean_x) * (y - weighted_mean_y)) denominator = np.sum(weights * (x - weighted_mean_x) ** 2) beta_1 = numerator / denominator
Intercept (beta_0)
beta_0 = weighted_mean_y - beta_1 * weighted_mean_x
beta_0, beta_1The weighted least-squares regression line is:
Would you like a detailed explanation of how the weights influenced the result, or assistance with a related question?
Related Questions:
- How does the weighting impact the regression line compared to an unweighted fit?
- Can you provide the unweighted least-squares solution for comparison?
- What are the implications of assigning different weights in a regression analysis?
- How would adding more data points change the regression line?
- What is the standard error of the regression line for this fit?
Tip:
In weighted least-squares regression, assigning appropriate weights can significantly improve the accuracy of your model when some data points are less reliable.
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Math Problem Analysis
Mathematical Concepts
Weighted least-squares regression
Linear algebra
Statistics
Formulas
Weighted mean: x̄_w = Σ(w_i * x_i) / Σw_i
Weighted slope: β1 = Σ(w_i * (x_i - x̄_w) * (y_i - ȳ_w)) / Σ(w_i * (x_i - x̄_w)^2)
Intercept: β0 = ȳ_w - β1 * x̄_w
Theorems
Weighted least-squares method
Suitable Grade Level
Undergraduate Statistics or High School Advanced Math
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