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 tenth. Using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 80.

Solution

The question in your image asks to find the linear regression equation between homework grades xx and test grades yy and then use that equation to predict the test grade when the homework grade is 80.

To do this, we will:

  1. Find the linear regression equation y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.
  2. Use the least-squares method to determine mm and bb, and round the coefficients to the nearest tenth.
  3. Plug in x=80x = 80 into the equation to find the corresponding test grade.

Step 1: Organize the Data

From the table:

\hline \text{Homework Grade} (x) & \text{Test Grade} (y) \\ \hline 71 & 63 \\ 72 & 75 \\ 62 & 58 \\ 80 & 77 \\ 85 & 71 \\ 74 & 65 \\ 90 & 95 \\ 83 & 68 \\ 62 & 57 \\ \hline \end{array}$$ ### Step 2: Calculate the Linear Regression Coefficients To get $$m$$ and $$b$$, we use the formulas for the slope and intercept: $$m = \frac{n \sum (xy) - \sum x \sum y}{n \sum (x^2) - (\sum x)^2}$$ $$b = \frac{\sum y - m \sum x}{n}$$ Where $$n$$ is the number of data points. I'll now calculate these values.The linear regression equation that represents the relationship between homework grade $$x$$ and test grade $$y$$ is: $$y = 0.96x - 2.8$$ Now, for a homework grade of 80, the projected test grade can be calculated as: $$y = 0.96(80) - 2.8 = 74.3$$ Rounding to the nearest integer, the projected test grade is approximately **74**. Would you like further details or have any questions about the process? Here are 5 related questions you might find useful: 1. What does the slope in this context represent? 2. How does the correlation between homework and test grades affect predictions? 3. Can linear regression always provide accurate predictions for all data types? 4. How would you calculate this manually step by step? 5. How does rounding coefficients impact the accuracy of predictions? **Tip:** When interpreting linear regression, the slope tells you how much the dependent variable (test grade) changes with each unit increase in the independent variable (homework grade).

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Algebra
Statistics

Formulas

Slope formula: m = (n Σ(xy) - Σx Σy) / (n Σ(x^2) - (Σx)^2)
Intercept formula: b = (Σy - m Σx) / n
Linear equation: y = mx + b

Theorems

Least Squares Method

Suitable Grade Level

Grades 9-12