Based on the data, calculate and plot the residuals.

Understand the Problem

The question is asking to calculate the residuals from the predicted new scores based on the original scores using the given regression line. This involves applying the regression equation to each original score, determining the predicted new score, and then calculating the residual (the difference between the actual new score and the predicted new score). Finally, it requests plotting these residuals.

Answer

The residuals are: 0.1, -0.5, 0.5, -0.3, 0.1.

Steps to Solve

Apply the regression equation

Use the regression equation ( y = 0.8x + 4.3 ) to predict the new score for each original score.

For an original score of 2: $$ y = 0.8(2) + 4.3 = 1.6 + 4.3 = 5.9 $$
For an original score of 4: $$ y = 0.8(4) + 4.3 = 3.2 + 4.3 = 7.5 $$
For another original score of 4: $$ y = 0.8(4) + 4.3 = 3.2 + 4.3 = 7.5 $$
For an original score of 5: $$ y = 0.8(5) + 4.3 = 4 + 4.3 = 8.3 $$
For an original score of 7: $$ y = 0.8(7) + 4.3 = 5.6 + 4.3 = 9.9 $$

Calculate the residuals

The residual is calculated by subtracting the predicted score from the actual new score.

For the original score of 2: $$ \text{Residual} = 6 - 5.9 = 0.1 $$
For the original score of 4: $$ \text{Residual} = 7 - 7.5 = -0.5 $$
For the second original score of 4: $$ \text{Residual} = 8 - 7.5 = 0.5 $$
For the original score of 5: $$ \text{Residual} = 8 - 8.3 = -0.3 $$
For the original score of 7: $$ \text{Residual} = 10 - 9.9 = 0.1 $$

Summarize the results

Now that we have the residuals, we can summarize them:

For original score 2: Residual = 0.1
For original score 4: Residual = -0.5
For the second original score 4: Residual = 0.5
For original score 5: Residual = -0.3
For original score 7: Residual = 0.1

The calculated residuals for the original scores are:

Original score 2: Residual = 0.1
Original score 4: Residual = -0.5
Original score 4: Residual = 0.5
Original score 5: Residual = -0.3
Original score 7: Residual = 0.1

More Information

Residuals represent the differences between the actual new scores and the predicted new scores from the regression line. They indicate how well the regression model predicts each student's new score. Positive residuals mean the model underestimated the new score, while negative residuals indicate overestimation.

Tips

Miscalculating predicted scores: Ensure that calculations using the regression equation are accurate!
Incorrectly determining residuals: Remember that residuals are the difference between the actual score and the predicted score.

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