A study was done to compare the speed x (in miles per hour)...

A study was done to compare the speed $x$ (in miles per hour) with the mileage $y$ (in miles per gallon) of an automobile. The results are shown in the table. (Source: Federal Highway Administration) $$ \begin{array}{|c|c|} \hline \text { Speed, } \boldsymbol{x} & \text { Mileage, } \boldsymbol{y} \\ \hline 15 & 22.3 \\ 20 & 25.5 \\ 25 & 27.5 \\ 30 & 29.0 \\ 35 & 28.8 \\ 40 & 30.0 \\ 45 & 29.9 \\ 50 & 30.2 \\ 55 & 30.4 \\ 60 & 28.8 \\ 65 & 27.4 \\ 70 & 25.3 \\ 75 & 23.3 \\ \hline \end{array} $$ (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of a graphing utility to find a quadratic model for the data. (c) Use a graphing utility to graph the model in the same viewing window as the scatter plot. (d) Estimate the speed for which the miles per gallon is greatest.

A study was done to compare the speed $x$ (in miles per hour) with the mileage $y$ (in miles per gallon) of an automobile. The results are shown in the table. (Source: Federal Highway Administration)
$$
\begin{array}{|c|c|}
\hline \text { Speed, } \boldsymbol{x} & \text { Mileage, } \boldsymbol{y} \\
\hline 15 & 22.3 \\
20 & 25.5 \\
25 & 27.5 \\
30 & 29.0 \\
35 & 28.8 \\
40 & 30.0 \\
45 & 29.9 \\
50 & 30.2 \\
55 & 30.4 \\
60 & 28.8 \\
65 & 27.4 \\
70 & 25.3 \\
75 & 23.3 \\
\hline
\end{array}
$$
(a) Use a graphing utility to create a scatter plot of the data.
(b) Use the regression feature of a graphing utility to find a quadratic model for the data.
(c) Use a graphing utility to graph the model in the same viewing window as the scatter plot.
(d) Estimate the speed for which the miles per gallon is greatest.

Key Concepts

Graphing Utility

A graphing utility is a computational tool, such as a graphing calculator or software, that allows users to plot data, fit regression models, and visualize functions. It helps in both exploratory data analysis and presentation of statistical models.

Quadratic Model

A quadratic model is a specific type of regression model that uses a second-degree polynomial function (of the form y = ax² + bx + c) to capture a curved relationship between the independent and dependent variables. It is particularly useful when the data demonstrates a parabolic trend.

Estimation of Extrema

In the context of a quadratic model, estimating the maximum or minimum value involves finding the vertex of the parabola. This vertex represents the point at which the function attains its highest or lowest value, and it is crucial for identifying optimal conditions in the modeled relationship.

Scatter Plot

A scatter plot is a graphical representation where each pair of data points is plotted as a dot on a two-dimensional coordinate system. It is used to visually examine the relationship or correlation between two quantitative variables.

Regression Analysis

Regression analysis is a statistical process for estimating the relationships among variables. It involves fitting a model to data to understand how the dependent variable changes when any one of the independent variables is varied.

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