Math Problem Statement

A mathematics teacher wanted to see the correlation between test scores and homework. The homework grade (x) and test grade (y) are given in the accompanying table. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest hundredth. Using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 64.

Solution

To solve this problem, we need to determine the linear regression equation of the form:

y=mx+by = mx + b

where:

  • mm is the slope of the line, calculated as: m=n(xy)(x)(y)n(x2)((x))2m = \frac{n \sum(xy) - \sum(x)\sum(y)}{n \sum(x^2) - (\sum(x))^2}
  • bb is the y-intercept, calculated as: b=(y)m(x)nb = \frac{\sum(y) - m \sum(x)}{n}

Steps:

  1. Extract values from the table:

    • Homework grades (xx): 84,71,76,80,89,72,87,86,5284, 71, 76, 80, 89, 72, 87, 86, 52
    • Test grades (yy): 72,59,71,76,82,68,88,89,5072, 59, 71, 76, 82, 68, 88, 89, 50
  2. Compute required values:

    • (x)\sum(x), (y)\sum(y), (x2)\sum(x^2), (y2)\sum(y^2), and (xy)\sum(xy)
    • Substitute these into the formulas for mm and bb.
  3. Predict the test grade for x=64x = 64.

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:

y=1.02x6.31y = 1.02x - 6.31

Using this equation, the projected test grade for a student with a homework grade of 6464 is approximately:

5959

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:

  1. How is the slope of the regression line interpreted in this context?
  2. What does the y-intercept represent in the equation?
  3. What is the correlation coefficient (rr) for this dataset, and what does it signify?
  4. How would the prediction change for a homework grade of 9090?
  5. 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., r2r^2) 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