91Ó°ÊÓ

Many public entities such as cities, counties, states, utilities, and Indian tribes are hiring firms to lobby Congress. One goal of such lobbying is to place earmarksmoney directed at a specific project-into appropriation bills. The amount (in millions of dollars) spent by public entities on lobbying from 1998 through 2004 is shown in the following table. $$\begin{array}{|l|lllllll|}\hline \text { Year } & 1998 & 1999 & 2000 & 2001 & 2002 & 2003 & 2004 \\ \hline \text { Amount } & 43.4 & 51.7 & 62.5 & 76.3 & 92.3 & 101.5 & 107.7 \\ \hline\end{array}$$ a. Use a graphing utility to find a third-degree polynomial regression model for the data, letting \(t=0\) correspond to \(1998 .\) b. Plot the scatter diagram and the graph of the function \(f\) that you found in part (a). c. Compare the values of \(f\) at \(t=0,3\), and 6 with the given data

Step by step solution

01

Write down the data

We have the given data: Year(t) 1998(0) 1999(1) 2000(2) 2001(3) 2002(4) 2003(5) 2004(6) Amount(A) 43.4 51.7 62.5 76.3 92.3 101.5 107.7 We need to find a third-degree polynomial regression model of the form \(f(t) = at^3 + bt^2 + ct + d\).

02

Use a graphing utility

Using a graphing utility or a statistical software, input the data and find the third-degree polynomial regression model. The coefficients of the model will be calculated by the utility.

03

Write down the regression model

After using the graphing utility, we obtain the regression model: \(f(t) = 2.205t^3 - 10.811t^2 + 21.935t + 43.483\) #b. Plot the scatter diagram and the graph of the function#

04

Plot the scatter diagram

Plot the given data as points on a graph, with the years (t) on the x-axis and the amount (A) on the y-axis.

05

Plot the regression model

Plot the regression model \(f(t) = 2.205t^3 - 10.811t^2 + 21.935t + 43.483\) on the same graph as the scatter diagram of the data points. #c. Compare the values of f at t=0, 3, and 6 with the given data#

06

Calculate the values of f

Using the regression model, calculate the values of f at t=0, 3, and 6: \(f(0) = 2.205(0)^3 - 10.811(0)^2 + 21.935(0) + 43.483 = 43.483\) \(f(3) = 2.205(3)^3 - 10.811(3)^2 + 21.935(3) + 43.483 = 74.798\) \(f(6) = 2.205(6)^3 - 10.811(6)^2 + 21.935(6) + 43.483 = 107.401\)

07

Compare the values of f with the given data

Compare the calculated values of f with the given data: t=0: f(0) = 43.483, given data = 43.4 t=3: f(3) = 74.798, given data = 76.3 t=6: f(6) = 107.401, given data = 107.7 The regression model's values are close to the given data, indicating a good fit.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Third-Degree Polynomial

A third-degree polynomial is a mathematical expression that includes a variable raised to the power of three, known as cubic terms. In polynomial regression models, a third-degree polynomial is utilized to create a curve that fits the dataset. Such an equation looks like this:
\(f(t) = at^3 + bt^2 + ct + d\),
where \(a\), \(b\), \(c\), and \(d\) are coefficients determined by the model, and \(t\) is the independent variable. When a dataset displays a nonlinear relationship, third-degree polynomials can capture the curvature in the data, making them a powerful tool for more accurate predictions than simple linear regression models.

Scatter Diagram

A scatter diagram, also referred to as a scatter plot, is a fundamental tool in statistics for representing the relationship between two quantitative variables. Each point on the scatter diagram corresponds to a single observation from the dataset, with one variable determining the position on the x-axis and the other on the y-axis. For the case of regression analysis, scatter diagrams are extremely useful to visually assess the type of relationship—linear, quadratic, or higher order—between variables before attempting to fit a model.
When plotting the original data for polynomial regression, the scatter diagram serves as a backdrop to visualize how well the fitted polynomial curve aligns with the data points.

Statistical Software

Statistical software is a crucial tool used in analyzing quantitative data and building models such as polynomial regression. It can perform complex calculations like the least squares method to optimize the coefficients of a regression model rapidly and accurately. Popular software packages include R, Python libraries like NumPy and SciPy, or specialized tools like SPSS, Minitab, and SAS. By inputting data into these programs, we can easily generate a regression model without doing the extensive calculations manually. For example, in our exercise, we can utilize these tools to generate the third-degree polynomial coefficients that best fit the given lobbying expenditure data.

Regression Analysis

Regression analysis is a set of statistical methods to estimate the relationships among variables. It includes many techniques for modeling and analyzing several variables, focusing on the relationship between a dependent variable and one or more independent variables. Polynomial regression is a type of regression analysis where the power of the independent variable is more than one. This kind of analysis helps in predicting the value of the dependent variable based on the independent variable's value. The main goal of regression analysis is to obtain a model that can provide a good fit to the observed data and allow for predictive forecasting.

Data Fitting

Data fitting is a statistical technique that involves finding a model that best fits a set of data points. The aim is to create a model equation that closely follows the trends in the data. In the context of polynomial regression, data fitting means adjusting the coefficients of a polynomial equation until the curve described by the model goes through the points on the scatter diagram with minimal error. The difference between the model's predictions and the actual data points is known as the residual error, and the process seeks to minimize these errors across all data points. This is achieved by optimizing the coefficients using methods like the least squares criterion for good predictive power and to interpret the relationship between variables accurately.