The Number Of Cases Of A New Disease Can Be Modeled By The Quadratic Regression Equation $y = -2x^2 + 36x + 6$, Where $x$ Represents The Year.Which Is The Best Prediction For The Number Of New Cases In Year 15?A. 614 B. 96 C. 194

Introduction

In the field of epidemiology, understanding the spread of diseases is crucial for developing effective prevention and treatment strategies. One way to model the number of cases of a new disease is by using a quadratic regression equation. In this article, we will explore a quadratic regression model that represents the number of cases of a new disease over a period of years. We will use this model to predict the number of new cases in a specific year.

The Quadratic Regression Equation

The quadratic regression equation used to model the number of cases of the new disease is given by:

$$y = -2x^2 + 36x + 6$$

where $x$ represents the year. This equation is a quadratic function, which means that it has a parabolic shape. The coefficient of the $x^2$ term is negative, indicating that the number of cases decreases as the year increases.

Understanding the Quadratic Regression Equation

To understand the quadratic regression equation, let's break it down into its individual components. The equation has three terms:

• $-2x^2$: This term represents the quadratic component of the equation. The coefficient of $-2$ indicates that the number of cases decreases as the year increases.
• $36x$: This term represents the linear component of the equation. The coefficient of $36$ indicates that the number of cases increases as the year increases.
• $6$: This term represents the constant component of the equation. The value of $6$ indicates that there are $6$ cases in the first year.

Predicting the Number of New Cases in Year 15

Now that we have a quadratic regression equation that models the number of cases of the new disease, we can use it to predict the number of new cases in a specific year. Let's say we want to predict the number of new cases in year $15$. To do this, we need to substitute $x = 15$ into the quadratic regression equation.

$$y = -2(15)^2 + 36(15) + 6$$

$$y = -2(225) + 540 + 6$$

$$y = -450 + 540 + 6$$

$$y = 96$$

Therefore, the best prediction for the number of new cases in year $15$ is $96$.

Conclusion

In this article, we have explored a quadratic regression model that represents the number of cases of a new disease over a period of years. We have used this model to predict the number of new cases in a specific year. The quadratic regression equation used in this model is $y = -2x^2 + 36x + 6$, where $x$ represents the year. By substituting $x = 15$ into this equation, we have predicted that the number of new cases in year $15$ is $96$.

References

• [1] "Quadratic Regression." MathWorld, Wolfram Research, www.mathworld.wolfram.com/QuadraticRegression.html.
• [2] "Epidemiology." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/science/epidemiology.

Mathematical Derivations

Derivation of the Quadratic Regression Equation

The quadratic regression equation used in this article is $y = -2x^2 + 36x + 6$. This equation is derived from a quadratic function of the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

To derive the quadratic regression equation, we need to find the values of $a$, $b$, and $c$ that best fit the data. This can be done using a method called least squares.

Least Squares Method

The least squares method is a statistical technique used to find the best fit line or curve to a set of data. It involves minimizing the sum of the squared differences between the observed values and the predicted values.

To apply the least squares method, we need to define a function that represents the sum of the squared differences between the observed values and the predicted values. This function is called the sum of squares function.

Sum of Squares Function

The sum of squares function is defined as:

$$S = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$

where $y_i$ is the observed value, $\hat{y}_i$ is the predicted value, and $n$ is the number of data points.

To find the values of $a$, $b$, and $c$ that minimize the sum of squares function, we need to take the partial derivatives of $S$ with respect to $a$, $b$, and $c$ and set them equal to zero.

Partial Derivatives

The partial derivatives of $S$ with respect to $a$, $b$, and $c$ are:

$$\frac{\partial S}{\partial a} = -2 \sum_{i=1}^{n} x_i^2 (y_i - \hat{y}_i)$$

$$\frac{\partial S}{\partial b} = -2 \sum_{i=1}^{n} x_i (y_i - \hat{y}_i)$$

$$\frac{\partial S}{\partial c} = -2 \sum_{i=1}^{n} (y_i - \hat{y}_i)$$

Setting these partial derivatives equal to zero and solving for $a$, $b$, and $c$, we get:

$$a = \frac{n \sum_{i=1}^{n} x_i^2 \hat{y}_i - \sum_{i=1}^{n} x_i^2 y_i}{n \sum_{i=1}^{n} x_i^2 - (\sum_{i=1}^{n} x_i)^2}$$

$$b = \frac{n \sum_{i=1}^{n} x_i \hat{y}_i - \sum_{i=1}^{n} x_i y_i}{n \sum_{i=1}^{n} x_i^2 - (\sum_{i=1}^{n} x_i)^2}$$

$$c = \frac{\sum_{i=1}^{n} \hat{y}_i - n \sum_{i=1}^{n} x_i \hat{y}_i / \sum_{i=1}^{n} x_i^2}{n}$$

Substituting these values into the quadratic function $y = ax^2 + bx + c$, we get the quadratic regression equation:

$$y = -2x^2 + 36x + 6$$

$$$$

Introduction

In our previous article, we explored a quadratic regression model that represents the number of cases of a new disease over a period of years. We used this model to predict the number of new cases in a specific year. In this article, we will answer some frequently asked questions about the quadratic regression model.

Q: What is a quadratic regression model?

A: A quadratic regression model is a statistical model that uses a quadratic function to represent the relationship between a dependent variable and one or more independent variables. In our case, the dependent variable is the number of cases of a new disease, and the independent variable is the year.

Q: How is the quadratic regression model derived?

A: The quadratic regression model is derived using a method called least squares. This method involves minimizing the sum of the squared differences between the observed values and the predicted values.

Q: What is the sum of squares function?

A: The sum of squares function is a mathematical function that represents the sum of the squared differences between the observed values and the predicted values. It is defined as:

$$S = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$

where $y_i$ is the observed value, $\hat{y}_i$ is the predicted value, and $n$ is the number of data points.

Q: How is the quadratic regression equation derived?

A: The quadratic regression equation is derived by taking the partial derivatives of the sum of squares function with respect to the coefficients of the quadratic function and setting them equal to zero. The resulting equation is:

$$y = -2x^2 + 36x + 6$$

Q: What is the significance of the quadratic regression model?

A: The quadratic regression model is significant because it provides a mathematical representation of the relationship between the number of cases of a new disease and the year. This model can be used to predict the number of new cases in a specific year, which is useful for public health officials and policymakers.

Q: How can the quadratic regression model be used in real-world applications?

A: The quadratic regression model can be used in real-world applications such as:

• Predicting the number of cases of a new disease in a specific year
• Identifying the factors that contribute to the spread of a disease
• Developing effective prevention and treatment strategies
• Evaluating the effectiveness of public health interventions

Q: What are the limitations of the quadratic regression model?

A: The quadratic regression model has several limitations, including:

• It assumes a linear relationship between the dependent variable and the independent variable, which may not always be the case
• It is sensitive to outliers and data errors
• It may not capture non-linear relationships between the variables
• It requires a large sample size to be accurate

Conclusion

In this article, we have answered some frequently asked questions about the quadratic regression model. We have discussed the derivation of the model, its significance, and its limitations. We have also provided examples of how the model can be used in real-world applications.

References

• [1] "Quadratic Regression." MathWorld, Wolfram Research, www.mathworld.wolfram.com/QuadraticRegression.html.
• [2] "Epidemiology." Encyclopedia Britannica, Encyclopedia Britannica, Inc., www.britannica.com/science/epidemiology.

Mathematical Derivations

Derivation of the Quadratic Regression Equation

The quadratic regression equation used in this article is $y = -2x^2 + 36x + 6$. This equation is derived from a quadratic function of the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

To derive the quadratic regression equation, we need to find the values of $a$, $b$, and $c$ that best fit the data. This can be done using a method called least squares.

Least Squares Method

The least squares method is a statistical technique used to find the best fit line or curve to a set of data. It involves minimizing the sum of the squared differences between the observed values and the predicted values.

To apply the least squares method, we need to define a function that represents the sum of the squared differences between the observed values and the predicted values. This function is called the sum of squares function.

Sum of Squares Function

The sum of squares function is defined as:

$$S = \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$

where $y_i$ is the observed value, $\hat{y}_i$ is the predicted value, and $n$ is the number of data points.

To find the values of $a$, $b$, and $c$ that minimize the sum of squares function, we need to take the partial derivatives of $S$ with respect to $a$, $b$, and $c$ and set them equal to zero.

Partial Derivatives

The partial derivatives of $S$ with respect to $a$, $b$, and $c$ are:

$$\frac{\partial S}{\partial a} = -2 \sum_{i=1}^{n} x_i^2 (y_i - \hat{y}_i)$$

$$\frac{\partial S}{\partial b} = -2 \sum_{i=1}^{n} x_i (y_i - \hat{y}_i)$$

$$\frac{\partial S}{\partial c} = -2 \sum_{i=1}^{n} (y_i - \hat{y}_i)$$

Setting these partial derivatives equal to zero and solving for $a$, $b$, and $c$, we get:

$$a = \frac{n \sum_{i=1}^{n} x_i^2 \hat{y}_i - \sum_{i=1}^{n} x_i^2 y_i}{n \sum_{i=1}^{n} x_i^2 - (\sum_{i=1}^{n} x_i)^2}$$

$$b = \frac{n \sum_{i=1}^{n} x_i \hat{y}_i - \sum_{i=1}^{n} x_i y_i}{n \sum_{i=1}^{n} x_i^2 - (\sum_{i=1}^{n} x_i)^2}$$

$$c = \frac{\sum_{i=1}^{n} \hat{y}_i - n \sum_{i=1}^{n} x_i \hat{y}_i / \sum_{i=1}^{n} x_i^2}{n}$$

Substituting these values into the quadratic function $y = ax^2 + bx + c$, we get the quadratic regression equation:

$$y = -2x^2 + 36x + 6$$

This equation represents the number of cases of the new disease over a period of years. By substituting $x = 15$ into this equation, we can predict the number of new cases in year $15$.