The Number Of Cases Of A New Disease Can Be Modeled By The Quadratic Regression Equation $\hat{y} = -2x^2 + 40x + 8$, Where $x$ Represents The Year.Which Is The Best Prediction For The Number Of Cases In Year $15$?A. 276 B.

Introduction

Quadratic regression equations are a powerful tool in modeling real-world phenomena, including the number of cases of a new disease. In this article, we will analyze the quadratic regression equation $\hat{y} = -2x^2 + 40x + 8$, where $x$ represents the year, to determine the best prediction for the number of cases in year $15$.

Understanding the Quadratic Regression Equation

The quadratic regression equation $\hat{y} = -2x^2 + 40x + 8$ is a mathematical model that describes the relationship between the number of cases of a new disease and the year. The equation consists of three terms: a quadratic term $-2x^2$, a linear term $40x$, and a constant term $8$. The quadratic term represents the curvature of the relationship, while the linear term represents the rate of change of the relationship.

Interpreting the Coefficients

To understand the quadratic regression equation, we need to interpret the coefficients of each term. The coefficient of the quadratic term, $-2$, represents the rate of change of the relationship with respect to the year. A negative coefficient indicates that the relationship is decreasing over time. The coefficient of the linear term, $40$, represents the rate of change of the relationship with respect to the year. A positive coefficient indicates that the relationship is increasing over time. The constant term, $8$, represents the initial value of the relationship.

Predicting the Number of Cases in Year 15

To predict the number of cases in year $15$, we need to substitute $x = 15$ into the quadratic regression equation. This will give us the predicted value of the number of cases in year $15$.

$\hat{y} = -2(15)^2 + 40(15) + 8$

Calculating the Predicted Value

To calculate the predicted value, we need to follow the order of operations (PEMDAS):

1. Evaluate the exponent: $(15)^2 = 225$
2. Multiply the quadratic term: $-2(225) = -450$
3. Multiply the linear term: $40(15) = 600$
4. Add the constant term: $-450 + 600 + 8 = 158$

Conclusion

The predicted value of the number of cases in year $15$ is $158$. This means that according to the quadratic regression equation, the number of cases of the new disease is expected to be $158$ in year $15$.

Limitations of the Quadratic Regression Equation

While the quadratic regression equation provides a good fit to the data, it has some limitations. The equation assumes a linear relationship between the number of cases and the year, which may not be the case in reality. Additionally, the equation does not take into account other factors that may affect the number of cases, such as seasonality or external events.

Future Directions

Future research should focus on developing more accurate models that take into account the complexities of the real world. This may involve incorporating additional variables, such as seasonality or external events, into the model. Additionally, researchers should continue to collect and analyze data to improve the accuracy of the model.

References

• [1] "Quadratic Regression" by Math Is Fun. Retrieved February 26, 2024.
• [2] "Quadratic Equations" by Khan Academy. Retrieved February 26, 2024.

Discussion

The quadratic regression equation $\hat{y} = -2x^2 + 40x + 8$ provides a good fit to the data and predicts the number of cases in year $15$ to be $158$. However, the equation has some limitations, including the assumption of a linear relationship between the number of cases and the year. Future research should focus on developing more accurate models that take into account the complexities of the real world.

Final Answer

The final answer is: $\boxed{158}$

Introduction

In our previous article, we analyzed the quadratic regression equation $\hat{y} = -2x^2 + 40x + 8$ to determine the best prediction for the number of cases in year $15$. In this article, we will answer some frequently asked questions (FAQs) about the quadratic regression equation and its application to the number of cases of a new disease.

Q&A

Q: What is a quadratic regression equation?

A: A quadratic regression equation is a mathematical model that describes the relationship between a dependent variable (in this case, the number of cases of a new disease) and an independent variable (in this case, the year). The equation consists of three terms: a quadratic term, a linear term, and a constant term.

Q: What is the purpose of the quadratic regression equation in this context?

A: The purpose of the quadratic regression equation is to predict the number of cases of a new disease in a given year. By analyzing the equation, we can determine the rate of change of the relationship between the number of cases and the year.

Q: What is the coefficient of the quadratic term in the equation?

A: The coefficient of the quadratic term in the equation is $-2$. This represents the rate of change of the relationship with respect to the year. A negative coefficient indicates that the relationship is decreasing over time.

Q: What is the coefficient of the linear term in the equation?

A: The coefficient of the linear term in the equation is $40$. This represents the rate of change of the relationship with respect to the year. A positive coefficient indicates that the relationship is increasing over time.

Q: What is the constant term in the equation?

A: The constant term in the equation is $8$. This represents the initial value of the relationship.

Q: How do I calculate the predicted value of the number of cases in a given year?

A: To calculate the predicted value, you need to substitute the given year into the equation and follow the order of operations (PEMDAS).

Q: What are some limitations of the quadratic regression equation?

A: Some limitations of the quadratic regression equation include the assumption of a linear relationship between the number of cases and the year, and the failure to take into account other factors that may affect the number of cases, such as seasonality or external events.

Q: What are some future directions for research on this topic?

A: Future research should focus on developing more accurate models that take into account the complexities of the real world. This may involve incorporating additional variables, such as seasonality or external events, into the model.

Conclusion

The quadratic regression equation $\hat{y} = -2x^2 + 40x + 8$ provides a good fit to the data and predicts the number of cases in year $15$ to be $158$. However, the equation has some limitations, including the assumption of a linear relationship between the number of cases and the year. Future research should focus on developing more accurate models that take into account the complexities of the real world.

Final Answer

The final answer is: $\boxed{158}$

References

• [1] "Quadratic Regression" by Math Is Fun. Retrieved February 26, 2024.
• [2] "Quadratic Equations" by Khan Academy. Retrieved February 26, 2024.

Discussion

The quadratic regression equation $\hat{y} = -2x^2 + 40x + 8$ provides a good fit to the data and predicts the number of cases in year $15$ to be $158$. However, the equation has some limitations, including the assumption of a linear relationship between the number of cases and the year. Future research should focus on developing more accurate models that take into account the complexities of the real world.