Math Problem Statement

Interpret the slope of the regression line, predict the math score for a student with a verbal score of 453, and calculate the residual for a student with a perfect score of 1600.

Solution

Let's break down the information shown in the image step by step, related to a linear regression model connecting verbal scores and math scores.

Provided Regression Equation:

Math Score=196.368+0.640×(Verbal Score)\text{Math Score} = 196.368 + 0.640 \times (\text{Verbal Score})

  • Intercept (196.368): The predicted math score when the verbal score is 0.
  • Slope (0.640): For each additional point in verbal score, the model predicts an increase of 0.640 points in the math score.

Questions:

e) Interpretation of the Slope

The question asks you to interpret the slope of the line, which is 0.640.

  • Correct Choice (D): For each additional point in verbal score, the model predicts an increase of 0.640 points in the math score.

f) Prediction for a Verbal Score of 453

To predict the math score for a student with a verbal score of 453, substitute the verbal score into the equation: Math Score=196.368+0.640×453\text{Math Score} = 196.368 + 0.640 \times 453 Math Score=196.368+289.92=486.288\text{Math Score} = 196.368 + 289.92 = 486.288 Rounded to one decimal place: 486.4.

g) Residual for a Perfect Verbal Score of 1600

To find the residual, we first calculate the predicted math score for a verbal score of 1600: Math Score=196.368+0.640×1600=196.368+1024=1220.368\text{Math Score} = 196.368 + 0.640 \times 1600 = 196.368 + 1024 = 1220.368

If a student scores a perfect math score of 1600, the residual is calculated as: Residual=Actual ScorePredicted Score\text{Residual} = \text{Actual Score} - \text{Predicted Score} Residual=16001220.368=379.632\text{Residual} = 1600 - 1220.368 = 379.632 Rounded to one decimal place: 379.6.

Summary:

  • e): For each additional point in verbal score, the math score increases by 0.640 points.
  • f): Predicted math score for a verbal score of 453 is 486.4.
  • g): The residual for a student scoring 1600 on both verbal and math is 379.6.

Do you need further clarification on any of these steps? Here are 5 related questions to extend the topic:

  1. How does the R-squared value affect the accuracy of predictions in this model?
  2. What does the intercept of 196.368 represent in this context?
  3. How would the residual change if a student's actual math score was below the predicted value?
  4. What is the significance of the slope being positive in this regression model?
  5. How might this model differ if the relationship between verbal and math scores was non-linear?

Tip: When interpreting residuals, a positive residual means the actual value is higher than the predicted one, and a negative residual indicates the opposite.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Slope Interpretation
Residual Calculation

Formulas

Linear regression equation: y = mx + b
Residual = Actual value - Predicted value

Theorems

Linear Regression Theorem

Suitable Grade Level

Grades 9-12