At a Chandler-Gilbert Community College, an MAT 121 Intermediate Algebra course tracks student performance on a standardized exam over multiple semesters. The college records the average exam scores based on the number of weeks students spend preparing. The data is shown in the table below:
| Weeks of Preparation $(w)$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Average Exam Score $(S)$ | 65 | 72 | 78 | 82 | 84 | 82 | 78 |
a. Conceptual Questions (round all coefficients to two decimal places):
1) What is the quadratic regression equation that models the relationship between weeks of preparation and exam scores?
2) Based on the quadratic regression equation, what is the vertex of the function, and what does it represent in this context?
3) What is the [tex]$y$[/tex]-intercept of the quadratic model, and what does it suggest about a student's preparation for the exam?
4) Based on the model, how many weeks of preparation would result in the highest possible exam score?
5) If a student prepares for 8 weeks, what does the model predict their exam score will be? Show your calculations.
b. Application & Interpretation Questions (round to two decimal places):
6) Find the coefficient of determination $(R^2)$ for this regression and explain what does this tell you?
7) If a student prepares for 3.5 weeks, estimate their expected exam score using the quadratic model.
8) Suppose the college wants students to score at least $80 \%$ on average. According to the model, during what range of weeks will the average exam score be 80 or higher?
9) If a student prepares for 7 weeks, the model predicts a lower score than if they had prepared for 5 weeks. Why does this happen?
10) Could this model be used to predict exam scores for a student preparing more than 10 weeks? Why or why not?
Answer (1)
To address the student's question, we need to create a quadratic regression model from the given data and answer several related questions.
Let's begin by examining each part:
a. Conceptual Questions (round all coefficients to two decimal places):
Quadratic Regression Equation: To find the quadratic regression equation, we can use statistical software or a calculator with regression capabilities. Using the data, the quadratic regression equation can be expressed as: S ( w ) = − 1.71 w 2 + 14.14 w + 52.86 Here, S ( w ) represents the average exam score based on w weeks of preparation.
Vertex of the Function: To find the vertex, we use the formula w = − 2 a b , where a = − 1.71 and b = 14.14 :
w = − 2 × − 1.71 14.14 ≈ 4.13 The vertex is approximately ( 4.13 , 84.15 ) , meaning that at about 4.13 weeks of preparation, the predicted score is highest at approximately 84.15. This represents the optimal preparation time for the highest score based on this model.
y -Intercept: The y -intercept occurs when w = 0 :
S ( 0 ) = 52.86 This suggests that if a student does not prepare (0 weeks), their expected score is 52.86.
Weeks for Highest Score: Based on the vertex, the model predicts the highest possible exam score will occur at approximately 4.13 weeks of preparation.
Predicted Score for 8 Weeks: Substitute w = 8 into the regression equation: S ( 8 ) = − 1.71 ( 8 ) 2 + 14.14 ( 8 ) + 52.86 S ( 8 ) = − 109.44 + 113.12 + 52.86 = 56.54 Therefore, the predicted score after 8 weeks of preparation is approximately 56.54.
b. Application & Interpretation Questions (round to two decimal places):
Coefficient of Determination R 2 : This value can be obtained from regression analysis tools. It tells us how well the model fits the data. If R 2 = 0.84 , it indicates that 84% of the variation in exam scores is explained by the weeks of preparation, showing a good fit.
Score for 3.5 Weeks of Preparation: Substitute w = 3.5 into the equation: S ( 3.5 ) = − 1.71 ( 3.5 ) 2 + 14.14 ( 3.5 ) + 52.86 S ( 3.5 ) = − 20.98 + 49.49 + 52.86 = 81.37 A student preparing for 3.5 weeks is predicted to score approximately 81.37.
Weeks for Score of 80 or Higher: We solve for w when S ( w ) ≥ 80 :
The equation to solve is − 1.71 w 2 + 14.14 w + 52.86 = 80 .
Solve this to get a range of w values within which the score is 80 or higher, approximately between 3.05 and 5.47 weeks.
Lower Score for 7 Weeks of Preparation: The quadratic model shows a decreasing trend after the vertex (4.13 weeks), indicating that too many weeks of preparation can lead to fatigue or other factors reducing performance.
Predicting Beyond 10 Weeks: The model is based on the data provided, typically captured up to the given maximum week (7 weeks). Predicting scores for significantly more than 10 weeks might not be accurate due to extrapolation beyond the data range, as outside factors not modeled could influence scores.
