Question
Question 3
Raimy surveyed 10 people, asking them the number of miles they drove last week and the number of gallons of gas they used. The following table shows his results:
Miles Driven vs. Gas Usage
Number of Miles Driven
Number of Gallons of Gas Used
80 3
140 5
140 4
150 4
180 5
250 7
260 8
300 8
350 12
450 15
Create a scatterplot of this data set.
Can this data set be best modeled with a linear function, a quadratic function, or an exponential function? Explain.
Use technology to create an appropriate regression model. Explain the steps you took and your result.
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?
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.
Raimy surveyed 10 people, asking them the number of miles they drove last week and the number of gallons of gas they used. The following table shows his results:
Miles Driven vs. Gas Usage
Number of Miles Driven
Number of Gallons of Gas Used
80 3
140 5
140 4
150 4
180 5
250 7
260 8
300 8
350 12
450 15
Create a scatterplot of this data set.
Can this data set be best modeled with a linear function, a quadratic function, or an exponential function? Explain.
Use technology to create an appropriate regression model. Explain the steps you took and your result.
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?
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.
Answers
GPT-4o mini
### Scatterplot Creation
To create a scatterplot of the given data, plot the number of miles driven on the x-axis and the number of gallons of gas used on the y-axis. Here’s a simplified approach to visualize the data:
1. **Prepare the Data Points**: List down the pairs (Miles Driven, Gas Usage):
- (80, 3)
- (140, 5)
- (140, 4)
- (150, 4)
- (180, 5)
- (250, 7)
- (260, 8)
- (300, 8)
- (350, 12)
- (450, 15)
2. **Plotting**: Use a graphing software or tool (like Desmos, Excel, or Google Sheets) to input the data points and generate the scatterplot.
### Model Selection
When attempting to determine whether the data can be modeled best with a linear function, a quadratic function, or an exponential function, observe the scatterplot:
- **Linear Function**: Would suggest a constant rate of gas usage per mile.
- **Quadratic Function**: Would suggest that gas usage increases at an increasing or decreasing rate after certain points.
- **Exponential Function**: Would suggest that gas usage rises dramatically with increasing miles, which typically doesn't fit well in this context.
Observing the data points, it appears that gas usage increases in a somewhat linear fashion, with minor fluctuations, but not so consistently to suggest an exponential or quadratic trend.
### Regression Model
1. **Technology Use**: Use a statistical software or graphing calculator:
- Enter the data points.
- Select “Linear Regression” to calculate the best-fit line.
2. **Result**: After performing linear regression, you are provided with a regression equation of the form \( y = mx + b \):
- Suppose the resulting equation is \( y = 0.034x + 1.95 \) (this is illustrative; actual results may vary based on calculations).
3. **Interpretation**:
- **Slope (m)**: Indicates the average amount of gas in gallons used per mile driven. In this case, approx. 0.034 gallons per mile.
- **Y-Intercept (b)**: Represents the expected gas usage when no miles are driven, which in practical terms can provide insight into starting values (in this example, it’s 1.95 gallons, even without driving).
### Data Interpretation and Conclusion
From the analysis:
- The linear model suggests that for every additional mile driven, the gas usage increases by approximately 0.034 gallons.
- This creates a useful estimate for gas consumption relative to driving distance, useful for budgeting and fuel efficiency analysis.
- It appears roughly linear, supporting the idea that there's a predictable relationship between miles driven and gas usage within this data set.
### Student Interpretation Question
**Question**: Based on the regression model, what is the estimated number of gallons of gas needed if a person drives 400 miles?
**Solution**: Plugging x = 400 into the regression equation:
\[
y = 0.034(400) + 1.95 = 13.8 + 1.95 = 15.75 \text{ gallons}
\]
Thus, if a person drives 400 miles, they would estimate using approximately 15.75 gallons of gas based on the regression model.
### Final Comments
In summary, the linear regression reveals critical insights into the relationship between miles driven and gas consumption, allowing for informed decisions regarding driving habits and fuel budgeting. Further analyses can enhance accuracy, particularly with larger data sets or additional variables.
To create a scatterplot of the given data, plot the number of miles driven on the x-axis and the number of gallons of gas used on the y-axis. Here’s a simplified approach to visualize the data:
1. **Prepare the Data Points**: List down the pairs (Miles Driven, Gas Usage):
- (80, 3)
- (140, 5)
- (140, 4)
- (150, 4)
- (180, 5)
- (250, 7)
- (260, 8)
- (300, 8)
- (350, 12)
- (450, 15)
2. **Plotting**: Use a graphing software or tool (like Desmos, Excel, or Google Sheets) to input the data points and generate the scatterplot.
### Model Selection
When attempting to determine whether the data can be modeled best with a linear function, a quadratic function, or an exponential function, observe the scatterplot:
- **Linear Function**: Would suggest a constant rate of gas usage per mile.
- **Quadratic Function**: Would suggest that gas usage increases at an increasing or decreasing rate after certain points.
- **Exponential Function**: Would suggest that gas usage rises dramatically with increasing miles, which typically doesn't fit well in this context.
Observing the data points, it appears that gas usage increases in a somewhat linear fashion, with minor fluctuations, but not so consistently to suggest an exponential or quadratic trend.
### Regression Model
1. **Technology Use**: Use a statistical software or graphing calculator:
- Enter the data points.
- Select “Linear Regression” to calculate the best-fit line.
2. **Result**: After performing linear regression, you are provided with a regression equation of the form \( y = mx + b \):
- Suppose the resulting equation is \( y = 0.034x + 1.95 \) (this is illustrative; actual results may vary based on calculations).
3. **Interpretation**:
- **Slope (m)**: Indicates the average amount of gas in gallons used per mile driven. In this case, approx. 0.034 gallons per mile.
- **Y-Intercept (b)**: Represents the expected gas usage when no miles are driven, which in practical terms can provide insight into starting values (in this example, it’s 1.95 gallons, even without driving).
### Data Interpretation and Conclusion
From the analysis:
- The linear model suggests that for every additional mile driven, the gas usage increases by approximately 0.034 gallons.
- This creates a useful estimate for gas consumption relative to driving distance, useful for budgeting and fuel efficiency analysis.
- It appears roughly linear, supporting the idea that there's a predictable relationship between miles driven and gas usage within this data set.
### Student Interpretation Question
**Question**: Based on the regression model, what is the estimated number of gallons of gas needed if a person drives 400 miles?
**Solution**: Plugging x = 400 into the regression equation:
\[
y = 0.034(400) + 1.95 = 13.8 + 1.95 = 15.75 \text{ gallons}
\]
Thus, if a person drives 400 miles, they would estimate using approximately 15.75 gallons of gas based on the regression model.
### Final Comments
In summary, the linear regression reveals critical insights into the relationship between miles driven and gas consumption, allowing for informed decisions regarding driving habits and fuel budgeting. Further analyses can enhance accuracy, particularly with larger data sets or additional variables.