Highway Design To allow enough distance for cars to pass on...

Highway Design To allow enough distance for cars to pass on two-lane highways, engineers calculate minimum sight distances between curves and hills. The table at top of the next page shows the minimum sight distance $y$ in feet for a car traveling at $x$ mph. (Image can't copy) $$\begin{array}{|c|c|c|c|c|} \hline x \text { (in mph) } & 20 & 30 & 40 & 50 \\ \hline y \text { (in feet) } & 810 & 1090 & 1480 & 1840 \end{array}$$ $$\begin{array}{|c|c|c|c} \hline x \text { (in mph) } & 60 & 65 & 70 \\ \hline y \text { (in feet) } & 2140 & 2310 & 2490 \end{array}$$ (a) Make a scatter diagram of the data. (b) Use the regression feature of a calculator to find the best-fitting linear function for the data. Graph the function with the data. (c) Repeat part (b) for a cubic function. (d) Use both functions from parts (b) and (c) to estimate the minimum sight distance for a car traveling 43 mph. (e) Which function fits the data better?

Highway Design To allow enough distance for cars to pass on two-lane highways, engineers calculate minimum sight distances between curves and hills. The table at top of the next page shows the minimum sight distance $y$ in feet for a car traveling at $x$ mph.
(Image can't copy)
$$\begin{array}{|c|c|c|c|c|}
\hline x \text { (in mph) } & 20 & 30 & 40 & 50 \\
\hline y \text { (in feet) } & 810 & 1090 & 1480 & 1840
\end{array}$$
$$\begin{array}{|c|c|c|c}
\hline x \text { (in mph) } & 60 & 65 & 70 \\
\hline y \text { (in feet) } & 2140 & 2310 & 2490
\end{array}$$
(a) Make a scatter diagram of the data.
(b) Use the regression feature of a calculator to find the best-fitting linear function for the data. Graph the function with the data.
(c) Repeat part (b) for a cubic function.
(d) Use both functions from parts (b) and (c) to estimate the minimum sight distance for a car traveling 43 mph.
(e) Which function fits the data better?

Key Concepts

Scatter Diagram

A scatter diagram is a graphical representation of paired numerical data, where each point on the graph represents a pair of values. This tool is used to visualize patterns, trends, or relationships between the two variables, making it easier to identify correlations or potential outliers in the data set.

Linear Regression

Linear regression is a statistical method for modeling the relationship between a dependent variable and one or more independent variables using a linear equation. It finds the best-fitting straight line through the data points by minimizing the sum of the squared differences between the observed values and those predicted by the model.

Cubic Regression

Cubic regression is an extension of polynomial regression that fits a third-degree polynomial to the data. It is useful for modeling more complex, nonlinear relationships by allowing the curve to change direction up to two times, potentially providing a better fit when the underlying data trend is not linear.

Model Prediction

Model prediction involves using a regression model to estimate the value of the dependent variable for a given value of the independent variable. This process is essential in many applications to make inferences or decisions based on the observed relationship between variables.

Model Comparison

Model comparison is the process of evaluating different regression models to determine which one best fits the data. This assessment typically involves comparing statistical measures such as the R-squared value, residual plots, or prediction errors, ensuring that the chosen model provides accurate predictions and reflects the underlying data trends.

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