91Ó°ÊÓ

The following table gives the number of miles per gallon in the city and on the highway for some of the most fuel efficient cars according to Consumer Reports. Make a scatterplot of the data using city mileage as the predictor variable. Find the regression equation and use it to predict the highway mileage for a fuel-efficient car that gets 40 miles per gallon in city driving. Would it be appropriate to use the regression equation to predict the highway mileage for a fuel-efficient car that got 60 miles per gallon in city driving? If so, make the prediction. If not, explain why it would be inappropriate to do so. $$ \begin{array}{|lcc|} \hline \text { Model } & \text { City Mileage } & \text { Highway Mileage } \\\ \hline \text { Toyota Prius 3 } & 43 & 59 \\ \hline \text { Hyundai Ioniq } & 42 & 60 \\ \hline \text { Toyota Prius Prime } & 38 & 62 \\ \hline \text { Kia Niro } & 33 & 52 \\ \hline \text { Toyota Prius C } & 37 & 48 \\ \hline \text { Chevrolet Malibu } & 33 & 49 \\ \hline \end{array} $$$$ \begin{array}{|lcc|} \hline \text { Model } & \text { City Mileage } & \text { Highway Mileage } \\\ \hline \text { Ford Fusion } & 35 & 41 \\ \hline \text { Hyundai Sonata } & 31 & 46 \\ \hline \text { Toyota Camry } & 32 & 43 \\ \hline \text { Ford C-Max } & 35 & 38 \\ \hline \end{array} $$

Step by step solution

01

Construct the scatterplot

Plot the city mileage (X-axis) and the highway mileage (Y-axis) of the vehicles in a scatterplot. This will give a visual representation of the relationship between the two variables.

02

Find the regression equation

From the scatterplot, use statistical software or calculator to compute for the least squares regression line, which will provide the regression equation.

03

Predict the highway mileage for 40 mpg city mileage

Plug 40, the city mileage, into the regression equation obtained from Step 2. The result would predict the highway mileage for a vehicle that has a city mileage of 40 mpg.

04

Assess the appropriateness of the model for 60 mpg city mileage

Check if 60 mpg for the city mileage is within the data range plotted in the scatterplot. If not, using the regression equation might be inappropriate because it involves extrapolation, and the prediction may not be reliable. Provide your reasons clearly.

05

Make the prediction or provide explanation

If 60 mpg is within the range, use the regression equation to predict the highway mileage. If it is out of range, explain why the regression equation may not give reliable prediction due to the extrapolation beyond the data.

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Key Concepts

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

Regression Equation

A **regression equation** is a mathematical formula that helps us understand the relationship between two variables. In this case, we are trying to relate city mileage to highway mileage using statistical methods.

The regression equation is typically derived using a method called the **least squares method**. This method finds the line that best fits the data points on your scatterplot. Essentially, it minimizes the sum of the squares of the differences between the observed values and the values predicted by the line.

A regression equation usually takes the form:

• \( y = mx + c \)

Here, \( y \) is the dependent variable (highway mileage), \( m \) is the slope of the line, \( x \) is the independent variable (city mileage), and \( c \) is the y-intercept.

By calculating the slope \( m \) and y-intercept \( c \), you create a tool to predict one variable from another. That's why once we have this equation, we can predict highway mileage for a given city mileage, like 40 miles per gallon, by plugging it into the equation.

Predictor Variable

The **predictor variable** is the variable that helps us make predictions about another variable in a mathematical model. For the exercise at hand, the city mileage acts as the predictor variable.

Predictor variables are plotted on the X-axis of a scatterplot. In our context, it means that we use the city mileage values to anticipate what the highway mileage will be for each car. This is why the city mileage data points sit along the horizontal line, guiding us to use them for predictions.

The choice of the predictor variable often depends on the natural relationship between variables and the context of the study. Here, it makes sense to assume city mileage could affect highway mileage or vice versa.

This setup allows us to analyze relationships effectively and identify any trends or patterns between city and highway mileage for the cars given, thereby helping in making accurate predictions using the regression equation.

Extrapolation

**Extrapolation** refers to the process of predicting or estimating values beyond the original observation range of the data set.

When we try to predict the highway mileage for a fuel-efficient car that gets 60 miles per gallon in city driving, as per this exercise, we encounter extrapolation. This is because the original data set only includes city mileage values up to 43 mpg. Predicting for a 60 mpg city mileage means going outside this proven range.

Extrapolation is often unreliable because the prediction relies on patterns presumed to continue beyond the observed data. In many cases, real-world behavior may change outside the original observation range, making predictions less reliable.

Therefore, while the scatterplot and regression equation are useful tools when confined within the original data range, caution should be exercised when applying these predictions to situations involving extrapolation. It highlights the importance of understanding your data limits and the potential risks of predictions that go beyond those limits.