To solve this problem, we will use the method of linear regression which involves finding the line of best fit for a set of data points. The linear regression equation has the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( m \)[/tex] is the slope of the line,
- [tex]\( b \)[/tex] is the y-intercept.
Given the data points:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Homework Grade (x)} & \text{Test Grade (y)} \\
\hline
72 & 75 \\
53 & 39 \\
90 & 81 \\
77 & 70 \\
82 & 83 \\
74 & 78 \\
78 & 75 \\
74 & 70 \\
\hline
\end{array}
\][/tex]
We will use the following formulas to find the slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex]:
[tex]\[ m = \frac{N (\sum xy) - (\sum x)(\sum y)}{N (\sum x^2) - (\sum x)^2} \][/tex]
[tex]\[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{N (\sum x^2) - (\sum x)^2} \][/tex]
where [tex]\( N \)[/tex] is the number of data points.
First, we calculate the required sums:
[tex]\[
\sum x = 72 + 53 + 90 + 77 + 82 + 74 + 78 + 74 = 600
\][/tex]
[tex]\[
\sum y = 75 + 39 + 81 + 70 + 83 + 78 + 75 + 70 = 591
\][/tex]
[tex]\[
\sum x^2 = 72^2 + 53^2 + 90^2 + 77^2 + 82^2 + 74^2 + 78^2 + 74^2 = 44919
\][/tex]
[tex]\[
\sum y^2 = 75^2 + 39^2 + 81^2 + 70^2 + 83^2 + 78^2 + 75^2 + 70^2 = 47165
\][/tex]
[tex]\[
\sum xy = 72 \cdot 75 + 53 \cdot 39 + 90 \cdot 81 + 77 \cdot 70 + 82 \cdot 83 + 74 \cdot 78 + 78 \cdot 75 + 74 \cdot 70 = 45687
\][/tex]
Now, substituting these values into the slope and intercept formulas:
[tex]\[
m = \frac{8 \cdot 45687 - 600 \cdot 591}{8 \cdot 44919 - 600^2} = \frac{365496 - 354600}{359352 - 360000} = \frac{10896}{-648} = -16.81
\][/tex]
[tex]\[
b = \frac{591 \cdot 44919 - 600 \cdot 45687}{8 \cdot 44919 - 600^2} = \frac{26524929 - 27412200}{359352 - 360000} = \frac{-887271}{-648} = 1369.22
\][/tex]
The linear regression equation:
[tex]\[ y = -16.81x + 1369.22 \][/tex]
To predict the test grade for a homework grade of 61, we substitute [tex]\( x = 61 \)[/tex]:
[tex]\[
y = -16.81(61) + 1369.22
\][/tex]
[tex]\[
y = -1025.41 + 1369.22
\][/tex]
[tex]\[
y = 343.81
\][/tex]
Rounded to the nearest integer, the predicted test grade is [tex]\( 344 \)[/tex].
So the predicted test grade for a student with a homework grade of 61 is [tex]\( 344 \)[/tex].
