To solve this, we will perform an exponential regression to model the data. The equation for exponential regression typically follows the form:

$y = a \cdot e^{b \cdot x}$

Where:

• $y$ is the debt (savings balance in this case),
• $x$ is the month,
• $a$ is the initial value when $x = 0$,
• $b$ is the rate of growth.

We will fit the data provided in the table to this model and use the resulting equation to predict the savings balance in month 11. Let me calculate this.import numpy as np from scipy.optimize import curve_fit

Given data

months = np.array([1, 2, 3, 4, 5, 6, 7, 8]) debt = np.array([620, 761.88, 899.80, 1039.93, 1270.63, 1589.04, 1851.31, 2154.92])

Define the exponential model: y = a * e^(b * x)

def exponential_model(x, a, b): return a * np.exp(b * x)

Fit the data to the model

params, _ = curve_fit(exponential_model, months, debt) a, b = params

Predict the balance for month 11

month_11_prediction = exponential_model(11, a, b)

month_11_prediction3724.706234469433724.70623446943successThe balance 11 months after graduation is predicted to be $3724.71.

Let me know if you’d like a detailed explanation of the exponential regression process or any other clarification.

Here are five related questions to expand on this:

1. How is exponential regression different from linear regression in modeling data?
2. What real-life scenarios can benefit from exponential models like this one?
3. How does the rate of growth ($b$) affect the predicted values?
4. Can we assess the accuracy of the model by calculating the residuals or $R^2$?
5. How would the prediction change if we used a logarithmic or polynomial model?

Tip: Exponential models are particularly useful for analyzing data with consistent percentage growth or decay over time.