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, estimate the homework grade, *to the nearest integer*, for a student with a test grade of 69.

Homework Grade (x)Test Grade (y)

7777

6464

7676

7676

6060

5959

7272

5757

7979

7070

7979

6464

7575

7777

7272

7777

To solve this problem, we need to calculate the linear regression equation for the given data. The general form of a linear regression equation is:

\[
y = mx + b
\]

where:
- \( y \) is the dependent variable (test grade),
- \( x \) is the independent variable (homework grade),
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.

### Step 1: Organize the data

We have the following pairs of values for \( x \) (homework grade) and \( y \) (test grade):

\[
(x, y) = \{(77, 77), (64, 64), (76, 76), (76, 76), (60, 60), (59, 59), (72, 72), (57, 57), (79, 79), (70, 70), (79, 79), (64, 64), (75, 75), (77, 77), (72, 72), (77, 77)\}
\]

### Step 2: Calculate the slope \( m \) and y-intercept \( b \)

The formulas for the slope and intercept are:

\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

\[
b = \frac{\sum y - m(\sum x)}{n}
\]

Where:
- \( n \) is the number of data points (16 in this case),
- \( \sum xy \) is the sum of the product of corresponding x and y values,
- \( \sum x \) is the sum of the x values,
- \( \sum y \) is the sum of the y values,
- \( \sum x^2 \) is the sum of the squares of the x values.

I'll calculate these values now.The linear regression equation representing the given data is:

\[
y = 1.0x + 0.0
\]

Using this equation, the estimated homework grade for a student with a test grade of 69 is:

\[
x = 69
\]

So, the homework grade estimate is **69**.

Would you like more details on this process or have any other questions?

Here are 5 related questions you might consider:
1. How do you calculate the correlation coefficient between homework and test scores?
2. What is the meaning of the slope in a linear regression model?
3. Can we interpret the regression equation differently for small datasets?
4. How would the equation change if an outlier was introduced in the data?
5. How can residuals help determine the fit of a regression model?

**Tip:** Always plot your data when doing linear regression to visually check the relationship!

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, estimate the homework grade, to the nearest integer, for a student with a test grade of 69.

Learn how to calculate the linear regression equation for homework and test scores. This problem demonstrates key statistical concepts such as the slope, intercept, and estimation using a real dataset. Find out how to estimate the homework grade for a student with a test score of 69. Suitable for Grades 10-12.

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