Number of Miles Driven

Question

80
140
140
150
180
250
260
300
350
450

Number of Gallons of Gas Used
3
5
4
4
5
7
8
8
12
15
A. Create a scatterplot of this data set. Include your scatterplot with your
submission.
B. Can this data set be best modeled with a linear function, a quadratic function, or an exponential function? Explain.
C. Use technology to create an appropriate regression model. Explain the steps you took and your result.
D. Use your scatterplot and regression model to interpret the data set in terms of the given context. What conclusions can you draw from this data set?
E. Write your own question that asks a student to interpret a key feature of the regression model in terms of the given context. Then, provide the solution to your question to a reasonable level of accuracy.

Answer 1

Let's address each part of your inquiry step by step.

A. Create a Scatterplot of This Data Set

To create a scatterplot, you need to plot the data points using the number of miles driven on the x-axis and the number of gallons of gas used on the y-axis. Since I'm unable to create visual content directly, I will guide you on how to do it using tools like Excel or Python (Matplotlib library).

Using Python (Matplotlib):

import matplotlib.pyplot as plt

miles = [80, 140, 140, 150, 180, 250, 260, 300, 350, 450]
gallons = [3, 5, 4, 4, 5, 7, 8, 8, 12, 15]

plt.scatter(miles, gallons)
plt.title('Scatterplot of Miles Driven vs. Gallons of Gas Used')
plt.xlabel('Miles Driven')
plt.ylabel('Gallons of Gas Used')
plt.grid()
plt.show()

Using Excel:

1. Enter the "Miles Driven" data in one column and "Gallons of Gas Used" in another column.
2. Highlight the data and select "Insert" > "Chart" > "Scatter" to create a scatterplot.

B. Can This Data Set Be Best Modeled with a Linear Function, a Quadratic Function, or an Exponential Function?

To determine the best model:

1. Visual Assessment: From the scatterplot, if the points roughly follow a straight line, a linear model may be appropriate. If they curve, a quadratic or exponential model could be better.

2. Fitting Models: You would typically fit linear, quadratic, and exponential regression models to evaluate their performance (R² values) and residuals.

Based on my evaluation, a linear model is likely a reasonable approximation, as the relationship appears to have a positive correlation. However, it may not hold for very high mileages, as gasoline consumption tends to level off.

C. Use Technology to Create an Appropriate Regression Model

Here's how to create a regression model using Python:

Using Python (NumPy and Statsmodels):

import numpy as np
import statsmodels.api as sm

# Convert data to numpy arrays
x = np.array(miles)
y = np.array(gallons)

# Add a constant to the independent variable (for intercept)
X = sm.add_constant(x)

# Fit the linear regression model
model = sm.OLS(y, X).fit()

# Print the summary of the regression
print(model.summary())

Using Excel:

1. Use the "Data Analysis Toolpak" and select "Regression".
2. Input the "Y Range" for Gallons and the "X Range" for Miles, and execute.

The output will provide the coefficients, R² value, and statistical significance.

D. Interpret the Data Set in Terms of the Given Context

Suppose the regression model showed a positive slope, indicating that as the number of miles driven increases, the gallons of gas used also tends to increase. The R² value will indicate how well the model explains the variance in gas usage.

From this data, we can conclude that driving more miles generally requires more gasoline. For instance, if the regression equation is \( Gallons = a + b \times Miles \), we can estimate the gallons required for any number of miles driven based on the model.

E. Draft a Question and Provide the Solution

Question: If a driver plans to drive 400 miles, how many gallons of gas can they expect to use according to the obtained regression model?

Solution:

1. Plug 400 into the regression equation \( Gallons = a + b \times 400 \).

2. Calculate the expected gallons based on the coefficients (a and b) from your regression output.

3. For example, if the linear model resulted in \( Gallons = 1 + 0.03 \times Miles \): \( Gallons = 1 + 0.03 \times 400 = 1 + 12 = 13 \) gallons.

4. Thus, the driver can expect to use approximately 13 gallons of gas for a 400-mile trip.

Use the actual coefficients derived from your regression analysis for precise computations.