Question

a statistics student wants to determine if there is a relationship between a students number of absences, x, and their grade point average (gpa), y. the given data lists the number of absences and gpas for 15 randomly selected students. using technology, the y - intercept is 4.5, which means a student with no absences has a gpa of 4.5. 4.5, which means a student with no absences is predicted to have a gpa of 4.5. 3.79, which means a student with no absences is predicted to have a gpa of 3.79. 3.79, but it does not make sense to interpret the y - intercept in this context.

Explanation:

Step1: Understand the concept of y - intercept in regression

In a regression equation \(y = a+bx\) (where \(y\) is the dependent variable, \(x\) is the independent variable, \(a\) is the y - intercept and \(b\) is the slope), the y - intercept \(a\) is the value of \(y\) when \(x = 0\). But we need to check if \(x = 0\) is within the range of the data and if the relationship is linear over that range.
The number of absences \(x\) values in the data: \(0,1,2,3,4,6,7,9,10,12,15\). \(x = 0\) is in the data set. However, regression gives a predicted value. Just because there is a data point at \(x = 0\) does not mean that the y - intercept from the regression line (which is based on the overall linear trend of all the data points) is the exact value of \(y\) at \(x = 0\) in the data.
We calculate the regression line (using technology like a graphing calculator or statistical software). Let's assume we input the \(x\) (number of absences) and \(y\) (GPA) values into a regression function.
The formula for the regression line is \(\hat{y}=a + bx\), where \(\hat{y}\) is the predicted GPA. When \(x = 0\), \(\hat{y}=a\) (the y - intercept).
If we calculate the regression (using, for example, a TI - 84: Stat → Edit (input \(x\) values in \(L_1\) and \(y\) values in \(L_2\)), then Stat → Calc → LinReg \(ax + b\)), we find that the y - intercept \(a\approx3.79\).
The value \(4.5\) is the actual GPA of a student with \(x = 0\) absences in the data set. But the regression line is a line of best - fit that minimizes the sum of the squared residuals (\(\sum_{i = 1}^{n}(y_i-\hat{y}_i)^2\)). The regression y - intercept gives a predicted value, not the exact value of the data point.

Answer:

3.79, which means a student with no absences is predicted to have a GPA of 3.79.