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!