Math Problem Statement
Solution
To solve this problem, we need to determine the linear regression equation of the form:
where:
- is the slope of the line, calculated as:
- is the y-intercept, calculated as:
Steps:
-
Extract values from the table:
- Homework grades ():
- Test grades ():
-
Compute required values:
- , , , , and
- Substitute these into the formulas for and .
-
Predict the test grade for .
Let me calculate these values. numpy as np
Data from the table
homework_grades = np.array([84, 71, 76, 80, 89, 72, 87, 86, 52]) # x values test_grades = np.array([72, 59, 71, 76, 82, 68, 88, 89, 50]) # y values
Number of data points
n = len(homework_grades)
Compute necessary sums for the formulas
sum_x = np.sum(homework_grades) sum_y = np.sum(test_grades) sum_x2 = np.sum(homework_grades**2) sum_xy = np.sum(homework_grades * test_grades)
Calculate slope (m) and intercept (b)
m = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x**2) b = (sum_y - m * sum_x) / n
Project the test grade for a homework grade of 64
predicted_grade = m * 64 + b
m, b, predicted_gradeThe linear regression equation is:
Using this equation, the projected test grade for a student with a homework grade of is approximately:
rounded to the nearest integer.
Would you like to see the detailed computations or graphs? Let me know if you have any questions!
Here are 5 related questions for further exploration:
- How is the slope of the regression line interpreted in this context?
- What does the y-intercept represent in the equation?
- What is the correlation coefficient () for this dataset, and what does it signify?
- How would the prediction change for a homework grade of ?
- Can this model be used to predict test grades accurately for students with homework grades far outside the given range?
Tip: Always evaluate the goodness of fit (e.g., ) of a regression model to ensure accurate predictions.
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Correlation
Algebra
Formulas
y = mx + b
m = (n * Σ(xy) - Σ(x)Σ(y)) / (n * Σ(x^2) - (Σ(x))^2)
b = (Σ(y) - m * Σ(x)) / n
Theorems
Linear Regression Slope-Intercept Form
Suitable Grade Level
Grades 10-12