To find the equation of the regression line using the given data, we will use the linear regression method. Given the dataset which contains the entering GPAs and the current GPAs of the students, we will follow these steps:
1. Calculate the means of the entering GPAs and current GPAs:
[tex]\[
\bar{X} = \text{mean of entering GPAs} = 3.67
\][/tex]
[tex]\[
\bar{Y} = \text{mean of current GPAs} = 3.75
\][/tex]
2. Determine the slope [tex]\( b_1 \)[/tex]:
We use the formula:
[tex]\[
b_1 = \frac{\sum{(X_i - \bar{X})(Y_i - \bar{Y})}}{\sum{(X_i - \bar{X})^2}}
\][/tex]
The summation of the numerator (covariance term) is:
[tex]\[
\sum{(X_i - \bar{X})(Y_i - \bar{Y})} = 0.010
\][/tex]
The summation of the denominator (variance term) is:
[tex]\[
\sum{(X_i - \bar{X})^2} = 0.193
\][/tex]
Therefore, the slope [tex]\( b_1 \)[/tex]:
[tex]\[
b_1 = \frac{0.010}{0.193} = 0.052
\][/tex]
3. Calculate the intercept [tex]\( b_0 \)[/tex]:
Using the formula:
[tex]\[
b_0 = \bar{Y} - b_1\bar{X}
\][/tex]
Substituting the values:
[tex]\[
b_0 = 3.75 - (0.052 \times 3.67) = 3.75 - 0.190 = 3.56
\][/tex]
4. Formulate the regression equation:
The equation of the regression line is given by:
[tex]\[
Y = b_0 + b_1X
\][/tex]
Substituting the values of the intercept and the slope:
[tex]\[
Y = 3.56 + 0.052X
\][/tex]
Thus, the equation of the regression line, rounded to three significant digits, is:
[tex]\[
Y = 3.56 + 0.052X
\][/tex]
