To find the linear regression equation, we need to use the formula for the line of best fit:

$y = mx + b$

Where:

• $m$ is the slope of the line, and
• $b$ is the y-intercept.

We can calculate the slope $m$ and y-intercept $b$ using the following formulas:

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

Where:

• $n$ is the number of data points.
• $\sum x$ is the sum of the x-values (homework grades).
• $\sum y$ is the sum of the y-values (test grades).
• $\sum xy$ is the sum of the products of the x and y values.
• $\sum x^2$ is the sum of the squares of the x-values.

Step 1: Calculate the necessary sums

Let's list out the data:

Homework (x)	Test Grade (y)	$x \times y$	$x^2$
71	63	71 * 63 = 4473	71² = 5041
72	75	72 * 75 = 5400	72² = 5184
62	58	62 * 58 = 3596	62² = 3844
80	77	80 * 77 = 6160	80² = 6400
85	71	85 * 71 = 6035	85² = 7225
74	65	74 * 65 = 4810	74² = 5476
90	95	90 * 95 = 8550	90² = 8100
83	68	83 * 68 = 5644	83² = 6889
62	57	62 * 57 = 3534	62² = 3844

Now let's compute the necessary sums:

• $\sum x = 71 + 72 + 62 + 80 + 85 + 74 + 90 + 83 + 62 = 719$
• $\sum y = 63 + 75 + 58 + 77 + 71 + 65 + 95 + 68 + 57 = 631$
• $\sum xy = 4473 + 5400 + 3596 + 6160 + 6035 + 4810 + 8550 + 5644 + 3534 = 44202$
• $\sum x^2 = 5041 + 5184 + 3844 + 6400 + 7225 + 5476 + 8100 + 6889 + 3844 = 49103$
• $n = 9$ (the number of data points).

Step 2: Compute the slope $m$

$m = \frac{9(44202) - (719)(631)}{9(49103) - (719)^2}$

$m = \frac{397818 - 453369}{441927 - 517761}$

$m = \frac{-55551}{-75834} \approx 0.73$

Step 3: Compute the y-intercept $b$

$b = \frac{\sum y - m \sum x}{n}$

$b = \frac{631 - (0.73)(719)}{9}$

$b = \frac{631 - 524.87}{9} \approx \frac{106.13}{9} \approx 11.8$

Step 4: Write the linear regression equation

The regression equation is approximately:

$y = 0.7x + 11.8$

Step 5: Find the projected test grade for a homework grade of 80

Now, we can plug $x = 80$ into the equation to find the projected test grade $y$:

$y = 0.7(80) + 11.8 = 56 + 11.8 = 67.8$

Rounding this to the nearest integer gives:

$y \approx 68$

Final Answer:

• Regression Equation: $y = 0.7x + 11.8$
• Projected Test Grade for Homework Grade of 80: 68

Would you like further details or clarifications? Here are some related questions:

1. How do you interpret the slope of the regression equation?
2. How does changing the number of data points affect the regression line?
3. What would happen if we used a nonlinear regression instead of a linear one?
4. How can we assess the accuracy of this regression equation?
5. How would you perform a hypothesis test on the slope of this regression line?

Tip: Always check if your data set suggests a linear relationship before applying linear regression.