Modeling Data The table shows the Consumer Price Index (CPI)...

Modeling Data The table shows the Consumer Price Index (CPI) for selected years. (Source: Bureau of Labor Statistics) $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \text { Year } & 1970 & 1975 & 1980 & 1985 & 1990 & 1995 & 2000 \\ \hline \text { CPI } & 38.8 & 53.8 & 82.4 & 107.6 & 130.7 & 152.4 & 172.2 \\ \hline \end{array} $$ (a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the CPI and $t$ represents the year, with $t=0$ corresponding to 1970 . (b) Use a graphing utility to plot the data and graph the model. Compare the data with the model. (c) Use the model to predict the CPI for the year 2010 .

Modeling Data The table shows the Consumer Price Index (CPI) for selected years. (Source: Bureau of Labor Statistics)
$$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline \text { Year } & 1970 & 1975 & 1980 & 1985 & 1990 & 1995 & 2000 \\
\hline \text { CPI } & 38.8 & 53.8 & 82.4 & 107.6 & 130.7 & 152.4 & 172.2 \\
\hline
\end{array}
$$
(a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the CPI and $t$ represents the year, with $t=0$ corresponding to 1970 .
(b) Use a graphing utility to plot the data and graph the model. Compare the data with the model.
(c) Use the model to predict the CPI for the year 2010 .

Key Concepts

Regression Analysis

Regression analysis is a statistical method used to estimate the relationships between a dependent variable and one or more independent variables. In the context of the given problem, regression helps in determining the coefficients of the quadratic model that best fits the historical CPI data, minimizing the error between the observed values and the predicted values of the model.

Quadratic Model

A quadratic model is a type of polynomial model represented by a second-degree equation of the form y = at² + bt + c. This model is particularly useful for capturing non-linear trends in data. In the given context, the quadratic model is used to approximate the behavior of the Consumer Price Index over time by fitting a curve that can approximate the observed relationship between time and CPI.

Graphing Utilities

Graphing utilities, including graphing calculators and computer software, are tools that facilitate the visualization and analysis of mathematical functions and data sets. They are instrumental in performing regression analysis, plotting data points, and graphing models, allowing users to visually compare the fit of a model to the actual data and make predictions based on the trends observed.

Prediction

Prediction involves using a model, once established, to estimate or forecast future values of a variable. In this problem, the quadratic model is used to predict the Consumer Price Index for the year 2010 based on the trend established by past data. Prediction is a key aspect of data analysis as it extends current understanding to anticipate future trends.

Data Modeling

Data modeling is the process of creating a structured representation of data, often using mathematical formulas or algorithms, to analyze underlying patterns and relationships. In the context of this problem, data modeling involves selecting a quadratic function to capture the trend of the CPI over time, which aids in understanding historical data and making informed predictions about future values.

Transcript

Please give Ace some feedback