
3.	(10 points) Create a scatter plot of mpg (dependent variable) against hp (independent variable) with the fitted linear regression line. 



4.	(15 points) Write down the equation of the linear regression line. Interpret the coefficient of determination (R-squared) in the context of this model.

5.	(10 points) Create a scatter plot of mpg (dependent variable) against hp (independent variable) with the fitted exponential regression curve. 

6.	(15 points) Write down the equation of the exponential regression curve. Interpret the coefficient of determination (R-squared) in the context of this model.

7.	(15 points) Compare the fitted linear and exponential regression models. Justify which model provides a better fit based on the coefficient of determination (R-squared).


To address your questions, let's walk through the necessary steps for each part. I’ll describe what would need to be done for each requirement, and if you have data to share, I can help generate and analyze it directly.

### 3. Scatter Plot with Linear Regression Line
To create this plot:

1. **Plot the scatter plot**: Plot `mpg` on the y-axis and `hp` on the x-axis.
2. **Fit the Linear Regression Line**: Use a linear regression model to calculate the line of best fit.
3. **Add the Line to the Plot**: Overlay this fitted line on the scatter plot to visually represent the relationship.

### 4. Equation of the Linear Regression Line and R-squared Interpretation
1. **Linear Regression Equation**: The equation of a linear regression line is generally given by:
   \[
   \text{mpg} = b_0 + b_1 \cdot \text{hp}
   \]
   where \( b_0 \) is the y-intercept and \( b_1 \) is the slope coefficient.

2. **Interpretation of R-squared**: The R-squared value represents the proportion of variance in `mpg` that is explained by `hp` in the model. A higher R-squared indicates a stronger linear relationship between `hp` and `mpg`.

### 5. Scatter Plot with Exponential Regression Curve
To create this plot:

1. **Fit an Exponential Regression Model**: Here, an exponential model would have the general form:
   \[
   \text{mpg} = a \cdot e^{b \cdot \text{hp}}
   \]
   where \( a \) and \( b \) are parameters to be estimated.

2. **Plot the Exponential Curve**: After fitting this model, overlay the fitted exponential curve on the scatter plot for comparison with the linear regression line.

### 6. Equation of the Exponential Regression Curve and R-squared Interpretation
1. **Exponential Regression Equation**: Write down the equation with estimated values for \( a \) and \( b \).
2. **Interpretation of R-squared**: As with the linear model, the R-squared for the exponential model indicates how well this model explains the variance in `mpg` based on `hp`.

### 7. Comparison of Linear and Exponential Models
1. **Compare R-squared Values**: The model with a higher R-squared generally provides a better fit to the data. 
2. **Justification**: If one model has a significantly higher R-squared, it suggests a better explanatory power. You could also consider the residuals to determine which model best fits the data.

If you have specific data for `mpg` and `hp`, please upload it, and I can generate these visualizations and calculations for you directly.

Would you like to provide the dataset, or would you like more detailed explanations on any of these steps?

### Additional Questions:
1. How do you calculate and interpret the slope and intercept in linear regression?
2. What other types of regression models could potentially fit the data better than linear or exponential?
3. Can R-squared values differ significantly for nonlinear models, and why?
4. How would you assess residuals to determine model fit?
5. Why is it useful to plot both models on the same graph for comparison?

#### Tip:
When analyzing the R-squared value, remember it’s possible for a model to fit data well without high R-squared if it captures essential patterns, especially in nonlinear relationships.

3. (10 points) Create a scatter plot of mpg (dependent variable) against hp (independent variable) with the fitted linear regression line. 

4. (15 points) Write down the equation of the linear regression line. Interpret the coefficient of determination (R-squared) in the context of this model.

5. (10 points) Create a scatter plot of mpg (dependent variable) against hp (independent variable) with the fitted exponential regression curve.

6. (15 points) Write down the equation of the exponential regression curve. Interpret the coefficient of determination (R-squared) in the context of this model.

7. (15 points) Compare the fitted linear and exponential regression models. Justify which model provides a better fit based on the coefficient of determination (R-squared).

Explore how to create scatter plots of mpg vs hp with linear and exponential regression models, and compare their effectiveness using the coefficient of determination (R-squared). Suitable for high school students studying regression analysis and model fitting.

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