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, estimate the homework grade, *to the nearest integer*, for a student with a test grade of 89.

Homework Grade (x)Test Grade (y)

7070

7575

7171

6161

7979

6969

6060

5454

8888

7878

9090

8181

7373

6262

8282

7575

5656

4747

To find the linear regression equation \( y = mx + b \) (where \( y \) represents the test grade and \( x \) represents the homework grade), we'll follow these steps:

### Step 1: Organize the Data

We are given the following pairs of \( x \) (homework grade) and \( y \) (test grade):

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
70 & 70 \\
75 & 75 \\
71 & 71 \\
61 & 61 \\
79 & 79 \\
69 & 69 \\
60 & 60 \\
54 & 54 \\
88 & 88 \\
78 & 78 \\
90 & 90 \\
81 & 81 \\
73 & 73 \\
62 & 62 \\
82 & 82 \\
75 & 75 \\
56 & 56 \\
47 & 47 \\
\hline
\end{array}
\]

### Step 2: Calculate Key Values

We need to compute the following sums and averages:

- \( \Sigma x \) (sum of homework grades),
- \( \Sigma y \) (sum of test grades),
- \( \Sigma xy \) (sum of products of homework and test grades),
- \( \Sigma x^2 \) (sum of squares of homework grades).

These values allow us to use the formulas for slope \( m \) and intercept \( b \):

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

Let me calculate these.The linear regression equation for the given data is:

\[
y = 1.00x + 0.00
\]

This means the test grade \( y \) is approximately equal to the homework grade \( x \). Now, to estimate the homework grade for a student with a test grade of 89, we set \( y = 89 \) and solve for \( x \):

\[
89 = 1.00x + 0.00 \implies x = 89
\]

Thus, the estimated homework grade for a student with a test grade of 89 is **89**.

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

Here are 5 related questions to explore:
1. What is the correlation coefficient for this data set?
2. How do we interpret a slope of 1 in this context?
3. Could outliers in the data affect the linear regression model? How?
4. What is the residual value for each data point using this regression equation?
5. How would the regression line change if more data points were added?

**Tip:** The slope \( m \) in a regression line indicates the rate of change of \( y \) with respect to \( x \). A slope of 1 suggests a direct 1:1 relationship between the two variables.

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, estimate the homework grade, to the nearest integer, for a student with a test grade of 89.

Learn how to calculate the linear regression equation for a dataset of homework and test grades. This problem involves determining the slope and intercept using the least squares method, and estimating the homework grade for a student with a test grade of 89. Suitable for students in Grades 10-12.

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