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 62.

To solve this problem, we will first find the linear regression equation that represents the relationship between the homework grade (\(x\)) and the test grade (\(y\)).

### Steps:

1. **Gather the data**: You provided a table of values, but since I don't have the exact numbers from the table, I will outline the general method. You can follow this procedure with the given data.

2. **Find the linear regression equation**: The general form of the linear regression equation is:
   \[
   y = mx + b
   \]
   where \(m\) is the slope, and \(b\) is the y-intercept. 

3. **Calculate the slope (\(m\))**:
   The slope is calculated using the formula:
   \[
   m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
   \]
   where:
   - \(n\) is the number of data points.
   - \(\sum xy\) is the sum of the products of \(x\) and \(y\) for each pair.
   - \(\sum x\) is the sum of all \(x\) values.
   - \(\sum y\) is the sum of all \(y\) values.
   - \(\sum x^2\) is the sum of the squares of all \(x\) values.

4. **Calculate the y-intercept (\(b\))**:
   The y-intercept is calculated using the formula:
   \[
   b = \frac{\sum y - m(\sum x)}{n}
   \]

5. **Use the regression equation to predict the test score**: Once you have the equation, substitute the homework grade of 62 into the equation to find the predicted test grade.

### Example Process (without exact numbers):

- Suppose after calculating, you find the slope \(m = 0.5\) and the intercept \(b = 25.6\), the regression equation would look like:
  \[
  y = 0.5x + 25.6
  \]

- To predict the test grade for a homework grade of 62:
  \[
  y = 0.5(62) + 25.6 = 31 + 25.6 = 56.6
  \]
  Rounding to the nearest integer, the projected test grade is **57**.

If you can provide the exact data points, I can calculate this more accurately. Would you like to upload the data or provide the values?

---

**Relative Questions**:
1. How is the slope (\(m\)) interpreted in a linear regression model?
2. What does the y-intercept (\(b\)) represent in this context?
3. How can you calculate the correlation coefficient (\(r\)) for this data?
4. How does the linear regression model change if the correlation is weak?
5. What assumptions are made when using linear regression?

**Tip**: Always check the residuals (the differences between actual and predicted values) to see how well the model fits the data!

Learn how to write a linear regression equation to determine the correlation between homework grades and test scores. This example demonstrates key statistical concepts, including how to calculate the slope and intercept of the regression line. Using the equation, predict the test grade for a student with a homework score of 62. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation