Math Problem Statement

A statistics professor would like to build a model relating student scores on the first test to the scores on the second test. The test scores from a random sample of 21 students who have previously taken the course are given in the table.

Test Scores Student First Test Grade Second Test Grade 1 61 71 2 43 80 3 50 73 4 95 54 5 46 81 6 67 73 7 58 75 8 69 71 9 96 55 10 53 72 11 72 63 12 67 73 13 46 73 14 80 67 15 63 66 16 44 76 17 55 75 18 77 68 19 93 61 20 86 60 21 52 79 Step 2 of 2 : Interpret the coefficient of the first test grade in the model.

Solution

To interpret the coefficient of the first test grade in a statistical model, we would typically be working with a linear regression model. Let's assume the professor is trying to establish the following model:

Second Test Grade=β0+β1×(First Test Grade)+ϵ\text{Second Test Grade} = \beta_0 + \beta_1 \times (\text{First Test Grade}) + \epsilon

Where:

  • β0\beta_0 is the intercept of the regression line (the predicted second test score when the first test score is zero).
  • β1\beta_1 is the slope or coefficient of the first test grade, which represents the change in the second test grade for each one-unit increase in the first test grade.
  • ϵ\epsilon is the error term, accounting for variability not explained by the model.

Interpreting β1\beta_1:

The coefficient β1\beta_1 tells us how much the second test grade is expected to increase or decrease for each point increase in the first test grade. In simpler terms:

  • If β1=2\beta_1 = 2, for example, it would mean that for each additional point on the first test, the second test grade is expected to increase by 2 points, on average.
  • If β1=1\beta_1 = -1, it would indicate that for each additional point on the first test, the second test grade would decrease by 1 point, on average.

To actually find β1\beta_1 (the coefficient), you would typically calculate it through the method of least squares or use a statistical software package to perform the regression analysis on the given data.

Would you like to go ahead and compute this coefficient for you, or is there something else you'd like to explore in detail?

Follow-up questions to expand on this:

  1. How do you perform a simple linear regression on a dataset like this?
  2. What does the intercept β0\beta_0 represent in this context?
  3. How can you assess the goodness of fit for this model (e.g., using R2R^2)?
  4. What assumptions are made in linear regression, and how might they apply to this dataset?
  5. How would you interpret the residuals of this model?

Tip: Always check the linearity of the relationship between the two variables before applying linear regression, as it assumes a straight-line relationship.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistical Modeling
Interpretation of Coefficients

Formulas

Second Test Grade = β₀ + β₁ × (First Test Grade) + ε

Theorems

Least Squares Method
Interpretation of Regression Coefficients

Suitable Grade Level

Grades 10-12