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 42.
Homework Grade (x)	Test Grade (y)
6161	4949
81	67
86	86
70	56
74	66
74	77
55	52
          
Regression Equation:
Final Answer:

To determine the linear regression equation that represents the relationship between the homework grade (\( x \)) and the test grade (\( y \)), we need to use the formula for the linear regression line, which is:

\[
y = mx + b
\]

Where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.

The slope \( m \) can be calculated using the formula:

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

And the y-intercept \( b \) is given by:

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

Where:
- \( n \) is the number of data points,
- \( \sum x \) is the sum of all x-values (homework grades),
- \( \sum y \) is the sum of all y-values (test grades),
- \( \sum xy \) is the sum of the products of corresponding x and y values,
- \( \sum x^2 \) is the sum of the squares of x-values.

### Step 1: Calculate the necessary sums

Let's start with the given dataTo determine the linear regression equation that represents the relationship between the homework grade (\( x \)) and the test grade (\( y \)), we need to use the formula for the linear regression line, which is:

\[
y = mx + b
\]

Where:
- \( m \) is the slope of the line,
- \( b \) is the y-intercept.

The slope \( m \) can be calculated using the formula:

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

And the y-intercept \( b \) is given by:

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

Where:
- \( n \) is the number of data points,
- \( \sum x \) is the sum of all x-values (homework grades),
- \( \sum y \) is the sum of all y-values (test grades),
- \( \sum xy \) is the sum of the products of corresponding x and y values,
- \( \sum x^2 \) is the sum of the squares of x-values.

### Step 1: Calculate the necessary sums

Let's start with the given 

| Homework Grade (x) | Test Grade (y) |
|--------------------|----------------|
| 61                 | 49             |
| 81                 | 67             |
| 86                 | 86             |
| 70                 | 56             |
| 74                 | 66             |
| 74                 | 77             |
| 55                 | 52             |

Now, let's calculate the required values:

\[
\sum x = 61 + 81 + 86 + 70 + 74 + 74 + 55
\]
\[
\sum y = 49 + 67 + 86 + 56 + 66 + 77 + 52
\]
\[
\sum xy = (61 \times 49) + (81 \times 67) + (86 \times 86) + (70 \times 56) + (74 \times 66) + (74 \times 77) + (55 \times 52)
\]
\[
\sum x^2 = 61^2 + 81^2 + 86^2 + 70^2 + 74^2 + 74^2 + 55^2
\]

Now, I'll calculate these.

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

Using the formulas provided earlier and rounding to the nearest tenth, I will calculate \( m \) and \( b \), and then write the regression equation.

### Step 3: Estimate the homework grade \( x \) for a test grade \( y = 42 \)

I will substitute \( y = 42 \) into the regression equation to estimate \( x \).

Let me compute these values.# Given data
homework_grades = [61, 81, 86, 70, 74, 74, 55]
test_grades = [49, 67, 86, 56, 66, 77, 52]

# Number of data points
n = len(homework_grades)

# Sum calculations
sum_x = sum(homework_grades)
sum_y = sum(test_grades)
sum_xy = sum(x * y for x, y in zip(homework_grades, test_grades))
sum_x_squared = sum(x**2 for x in homework_grades)

# Slope (m) calculation
m = (n * sum_xy - sum_x * sum_y) / (n * sum_x_squared - sum_x**2)

# Y-intercept (b) calculation
b = (sum_y - m * sum_x) / n

# Regression equation: y = mx + b
regression_eq = f"y = {m:.1f}x + {b:.1f}"

# Estimate homework grade (x) for test grade (y = 42)
test_grade = 42
estimated_homework_grade = (test_grade - b) / m

regression_eq, round(estimated_homework_grade)The linear regression equation that represents the given set of data is:

\[
y = 1.1x - 12.4
\]

Using this equation, the estimated homework grade for a student with a test grade of 42 is approximately **51**.

Do you have any questions or would you like more details on this? Here are five related questions to consider:

1. How is the slope of the regression line interpreted in the context of this data?
2. What is the significance of the y-intercept in this regression equation?
3. How would the regression equation change if one more data point was added?
4. What methods can be used to assess the accuracy of this regression model?
5. How does linear regression differ from other types of regression models?

**Tip:** When interpreting a regression equation, remember that the slope represents the average change in the dependent variable (y) for each unit increase in the independent variable (x).

Explore how to determine the linear regression equation that represents the relationship between homework grades and test grades. Learn how to estimate the homework grade for a given test grade using the linear regression model. Suitable for Grades 10-12.

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