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:
Where:
- is the intercept of the regression line (the predicted second test score when the first test score is zero).
- 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.
- is the error term, accounting for variability not explained by the model.
Interpreting :
The coefficient 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 , 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 , 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 (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:
- How do you perform a simple linear regression on a dataset like this?
- What does the intercept represent in this context?
- How can you assess the goodness of fit for this model (e.g., using )?
- What assumptions are made in linear regression, and how might they apply to this dataset?
- 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
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