GPA ACT
3.897 21
3.885 14
3.778 28
2.54 22
3.028 21
3.865 31
2.962 32
3.961 27
0.5 29
3.178 26
3.31 24
3.538 30
3.083 24
3.013 24
3.245 33
2.963 27
3.522 25
3.013 31
2.947 25
2.118 20
2.563 24
3.357 21
3.731 28
3.925 27
3.556 28
3.101 26
2.42 28
2.579 22
3.871 26
3.06 21
3.927 25
2.375 16
2.929 28
3.375 26
2.857 22
3.072 24
3.381 21
3.29 30
3.549 27
3.646 26
2.978 26
2.654 30
2.54 24
2.25 26
2.069 29
2.617 24
2.183 31

2 15
2.952 19
3.806 18
2.871 27
3.352 16
3.305 27
2.952 26
3.547 24
3.691 30
3.16 21
2.194 20
3.323 30
3.936 29
2.922 25
2.716 23
3.37 25
3.606 23
2.642 30
2.452 21
2.655 24
3.714 32
1.806 18
3.516 23
3.039 20
2.966 23
2.482 18
2.7 18
3.92 29
2.834 20
3.222 23
3.084 26
4 28
3.511 34
3.323 20
3.072 20
2.079 26
3.875 32
3.208 25
2.92 27
3.345 27
3.956 29
3.808 19
2.506 21
3.886 24
2.183 27
3.429 25
3.024 18
3.75 29

3.833 24
3.113 27
2.875 21
2.747 19
2.311 18
1.841 25
1.583 18
2.879 20
3.591 32
2.914 24
3.716 35
2.8 25
3.621 28
3.792 28
2.867 25
3.419 22
3.6 30
2.394 20
2.286 20
1.486 31
3.885 20
3.8 29
3.914 28
1.86 16
2.948 28

Obtain the residuals ei for the first five students. What does the sum of residuals for
those 120 students equals to

To solve this problem, we need to follow these steps:

1. **Linear Regression Model**: The residuals \(e_i\) are calculated by subtracting the predicted value from the actual value for each student, i.e., 
\[
e_i = y_i - \hat{y}_i
\]
where \(y_i\) is the actual value (ACT score), and \(\hat{y}_i\) is the predicted value of the ACT score based on the GPA using a linear regression model.

2. **Fitting the Model**: First, we need to fit a linear regression model using the GPA as the independent variable \(x_i\) and the ACT score as the dependent variable \(y_i\). This will give us the equation:
\[
\hat{y}_i = \beta_0 + \beta_1 x_i
\]
where \(\beta_0\) is the intercept and \(\beta_1\) is the slope of the regression line.

3. **Residuals for the First Five Students**: After fitting the model, we calculate the predicted values \(\hat{y}_i\) for the first five students and compute their residuals \(e_i = y_i - \hat{y}_i\).

4. **Sum of Residuals**: A key property of linear regression is that the sum of the residuals for the entire dataset is always zero. This is because the regression line minimizes the sum of squared residuals, and the sum of residuals themselves is zero by construction.

Let me calculate the linear regression line and the residuals for the first five students.It appears there was an issue because the sizes of the GPA and ACT arrays do not match, indicating a potential mismatch in the data points. I'll inspect the lengths of both arrays and ensure they are the same before proceeding. Let's resolve that.It looks like the GPA array has 120 data points, while the ACT array only has 109. To correct this, we need to either adjust or truncate the GPA array to match the length of the ACT array. I will truncate the GPA array to 109 entries and proceed with fitting the regression model and calculating the residuals.Here are the residuals for the first five students:

1. \( e_1 = -4.55 \)
2. \( e_2 = -11.54 \)
3. \( e_3 = 2.54 \)
4. \( e_4 = -2.57 \)
5. \( e_5 = -3.92 \)

As expected, the sum of the residuals for all 109 students is very close to zero, \( 9.24 \times 10^{-13} \), which is essentially zero due to rounding errors in the calculation.

Would you like more details on the regression model or the interpretation of residuals?

Here are some follow-up questions:
1. What does a positive residual indicate about the student's ACT score?
2. How would an outlier affect the residuals in a linear regression?
3. Why is the sum of residuals zero in linear regression?
4. How could we interpret the slope of the regression model?
5. What assumptions does linear regression make about the relationship between GPA and ACT scores?

**Tip**: Residuals can help identify students whose ACT performance was significantly different from what their GPA would predict.

Obtain the residuals ei for the first five students. What does the sum of residuals for those 120 students equals to?

Learn how to calculate the residuals for the first five students in a linear regression model using GPA and ACT scores. This problem covers key statistical concepts such as residuals and the linear regression equation. Understand how the sum of residuals equals zero in linear regression, a key property of the least squares estimation method.

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