(a) use a graphing utility to create a scatter plot of the...

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model. $$\begin{aligned} &(2,34.3), \quad(3,33.8), \quad(4,32.6), \quad(5,30.1), \quad(6,27.8)\\ &(7,22.5), \quad(8,19.1), \quad(9,14.8), \quad(10,9.4) . \quad(11,3.7)\\ &(12,-1.6) \end{aligned}$$

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model.
$$\begin{aligned}
&(2,34.3), \quad(3,33.8), \quad(4,32.6), \quad(5,30.1), \quad(6,27.8)\\
&(7,22.5), \quad(8,19.1), \quad(9,14.8), \quad(10,9.4) . \quad(11,3.7)\\
&(12,-1.6)
\end{aligned}$$

Key Concepts

Scatter Plot

A scatter plot is a type of data visualization that displays individual data points on two axes, allowing one to visually assess the relationship between two quantitative variables. It is useful for identifying patterns, trends, and potential outliers, which can indicate whether the relationship may be linear, quadratic, or follow another form.

Linear and Quadratic Models

These are specific types of mathematical models used in regression analysis to describe the relationship between variables. A linear model represents the relationship with a straight line, implying a constant rate of change, while a quadratic model includes a squared term, allowing it to capture curvature in the data. Deciding between these models depends on the pattern observed in the scatter plot.

Regression Analysis

Regression analysis is a statistical technique used to estimate the relationship between dependent and independent variables. It involves finding the best-fit function that minimizes the differences between the observed values and the values predicted by the model. This process helps in making predictions and understanding the underlying trend in the data.

Model Comparison and Residual Analysis

Once different models have been fitted to the data, it is important to compare their performance. Model comparison involves evaluating the goodness-of-fit, often by examining the residuals—the differences between the observed and predicted values. Analyzing these residuals helps determine how well the chosen model captures the behavior of the data.

Data Modeling Using Graphing Utilities

Graphing utilities provide interactive tools that enable users to create scatter plots, perform regression analyses, and visually compare the fitted models with the actual data. They simplify the process of model fitting and validation by offering features such as regression functions and the ability to overlay model graphs on the scatter plots.

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