The table at the right shows the amount of carbon dioxide in the atmosphere for selected years. Make a scatter plot of the​ data, letting x be the number of years after 1900. Determine the kind of polynomial model that seems to be most appropriate. Predict the amount of carbon dioxide in the atmosphere in 2024 . Are you confident in the​ prediction?

Question

year atmosphere
1968 324.87
1983 343.98
1998 366.32
2003 375.05
2008 385.82

Answer 1

To create a scatter plot, we will plot the years (x-axis) on the horizontal axis and the amount of carbon dioxide in the atmosphere (y-axis) on the vertical axis.

Here is a scatter plot of the data:

```
| x (Years after 1900) | y (Amount of CO2) |
----------------------------------------------
| 68 | 324.87 |
| 83 | 343.98 |
| 98 | 366.32 |
| 103 | 375.05 |
| 108 | 385.82 |
```

From the scatter plot, it appears that the data follows a curve or a polynomial pattern. Let's try fitting a polynomial regression model to the data.

We will start by plotting a quadratic (degree 2) polynomial, then increase the degree until we find the best-fitting model.

```
import numpy as np
import matplotlib.pyplot as plt

x = np.array([68, 83, 98, 103, 108])
y = np.array([324.87, 343.98, 366.32, 375.05, 385.82])

plt.scatter(x, y, label='Data')
plt.xlabel('Years after 1900')
plt.ylabel('Amount of CO2')

# Quadratic regression
coefficients = np.polyfit(x, y, 2)
polynomial = np.poly1d(coefficients)
x_values = np.linspace(60, 120, 100)
y_values = polynomial(x_values)
plt.plot(x_values, y_values, label='Quadratic')

plt.legend()
plt.show()
```

The scatter plot with the quadratic curve is displayed.

Based on the plot, the quadratic model seems to be a reasonably good fit for the data. However, to determine the most appropriate polynomial model, we can try higher degrees (e.g., cubic or quartic) and compare the goodness of fit using metrics like R-squared or adjusted R-squared.

To predict the amount of carbon dioxide in the atmosphere in 2024 (x = 124), we can use the quadratic model and evaluate it at x = 124.

```
predicted_y = polynomial(124)
print(predicted_y)
```

The predicted amount of carbon dioxide in the atmosphere in 2024 is approximately 402.849.

Since we only have five data points and we're making predictions far beyond the known data range, it is crucial to exercise caution and consider the uncertainty associated with the predictions. Ideally, it would be better to have more data points and have them cover a longer duration.