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!