To find the equation of the least-squares regression line for the given biology test scores, we will follow these steps:
1. Gather Data: We have two sets of scores, one for the first test and one for the second test.
- First test scores, [tex]\( x \)[/tex]: 44, 93, 70, 88, 81, 51, 76, 58, 42, 81, 67, 85
- Second test scores, [tex]\( y \)[/tex]: 52, 100, 69, 89, 79, 62, 80, 59, 50, 88, 69, 86
2. Find the Slope ( [tex]\( m \)[/tex] ) and Intercept ( [tex]\( b \)[/tex] ):
The slope [tex]\( m \)[/tex] and intercept [tex]\( b \)[/tex] of the regression line [tex]\( y = mx + b \)[/tex] are calculated using the least-squares method, which minimizes the sum of the squares of the vertical distances of the points from the line. The formulas for [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are as follows:
[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.
3. Plug in the Data and Calculate:
After performing the calculations:
- We find that the slope [tex]\( m \)[/tex] is approximately 0.889.
- We find that the intercept [tex]\( b \)[/tex] is approximately 11.667.
4. Form the Regression Line Equation:
With the slope and intercept, we can write the equation of the least-squares regression line as:
[tex]\[
y = 0.889x + 11.667
\][/tex]
Thus, the equation of the least-squares regression line, rounded to three decimal places, is:
[tex]\[
y = 0.889x + 11.667
\][/tex]
