Sure, let's go through each part of your request step by step.
A. Create a scatterplot of this data set.
To create a scatterplot, you would plot the Number of Miles Driven on the x-axis and the Number of Gallons of Gas Used on the y-axis. The points would be as follows:
- (80, 3)
- (140, 5)
- (140, 4)
- (150, 4)
- (180, 5)
- (250, 7)
- (260, 8)
- (300, 8)
- (350, 12)
- (450, 15)
Unfortunately, I cannot create graphical representations directly, but you can use software or graphing tools like Excel or Google Sheets to create the scatterplot by entering the data into a spreadsheet and selecting the scatter plot option.
B. Can this data set be best modeled with a linear function, a quadratic function, or an exponential function? Explain.
To determine the best model, we can analyze the scatterplot.
-
Linear Function: A linear model suggests that as miles driven increase, the gallons of gas used increases at a constant rate. In this case, since the points do not all form a straight line, a linear function may not be the best fit.
-
Quadratic Function: This type of function suggests that the relationship could be parabolic, which is not clearly depicted in the scatterplot, as the increase in gas used is somewhat consistent with increasing miles.
-
Exponential Function: An exponential function indicates that the gallons of gas used increases exponentially with miles driven. Given that the increase in gallons doesn't seem to be proportional throughout (i.e., larger increases in miles tend to cause larger increases in gallons), this might fit.
Based on visual inspection or regression analysis, if the points seem to be curving upwards consistently, an exponential function might be a suitable model.
C. Use technology to create an appropriate regression model. Explain the steps you took and your result.
Steps to create a regression model:
- Input Data: Enter the data into a statistical software or tool such as Excel, Google Sheets, R, Python, or a calculator with regression capabilities.
- Select Regression Type:
- If using Excel: a. Create a scatterplot chart. b. Use the "Add Trendline" feature, selecting "Exponential" (or "Linear" or "Polynomial" if you want to test those).
- View Results: The software will provide you with a regression equation and an R-squared value, indicating how well the model fits the data.
For instance, entering the data into Excel and choosing an exponential trendline might yield an equation like:
\[ y = a \cdot e^{(bx)} \] where \(y\) represents gallons and \(x\) where miles, with specific constants \(a\) and \(b\).
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?
Based on the scatterplot and regression model, we can observe a general trend that as the number of miles driven increases, the gallons of gas used also tends to increase. The regression model can quantify this relationship, allowing for predictions. For example, if we know someone plans to drive 300 miles, we can predict they will use around 8 gallons of gas (based on the regression equation).
This relationship suggests diminishing returns on fuel efficiency as distance increases. The scatterplot shows clustering at lower mile values and a spread at higher values, indicating variability in gas consumption possibly due to driving habits, vehicle types, or road conditions.
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.
Question: Based on the regression model you created, what would be the predicted number of gallons of gas used if one were to drive 400 miles?
Answer: Using the regression equation you derived, substitute \( x \) with 400 miles to calculate the predicted gallons of gas.
Assuming an example regression equation:
\[ y = 0.025e^{(0.005x)} \quad (this is a hypothetical function) \]
Substituting \( x = 400 \):
\[ y = 0.025e^{(0.005 \cdot 400)} = 0.025e^{2} \approx 0.025 \cdot 7.39 \approx 0.185 \]
In reality, use the specific regression model generated to find the accurate gallons of gas used.
Thus, if based on the hypothetical equation we assume a result of approximately 11 gallons, you could conclude that for a drive of 400 miles, one might expect to use around 11 gallons of gas, keeping in mind the limitations of the model and external factors influencing gas consumption.