Kace does not have a very good credit rating. In January of 2021, he found a lending institution that was willing to loan him $200,000 for 15 years at a higher than normal interest rate of 6% because of his poor credit rating. Examine the table of decreasing loan balances over the 15-year period. Use regression to determine a curve of best fit for this data.

Year	Interest	Principal	Ending Balance
1	$11,769.23	$8,483.33	$191,516.67
2	$11,246.00	$9,006.57	$182,510.10
3	$10,690.49	$9,562.07	$172,948.02
4	$10,100.72	$10,151.84	$162,796.18
5	$9,474.58	$10,777.98	$152,018.20
6	$8,809.82	$11,442.75	$140,575.45
7	$8,104.05	$12,148.51	$128,426.94
8	$7,354.76	$12,897.80	$115,529.13
9	$6,559.25	$13,693.31	$101,835.82
10	$5,714.68	$14,537.89	$87,297.94
11	$4,818.01	$15,434.55	$71,863.38
12	$3,866.04	$16,386.52	$55,476.86
13	$2,855.36	$17,397.21	$38,079.66
14	$1,782.34	$18,470.23	$19,609.43
15	$643.13	$19,609.43	$-0.00

To determine a curve of best fit for this loan balance data, we can apply regression analysis. Based on the nature of the data, where the loan balance is decreasing over time, an exponential decay model or a polynomial regression is appropriate. Let's begin by reviewing the possible models:

### Step 1: Analyze the Data
The data shows the loan balance decreasing year after year, which suggests the following types of models might fit:
- **Exponential decay**: This is common when the rate of change (in this case, balance reduction) decreases over time.
- **Polynomial**: A higher-order polynomial might also be a good fit if the relationship between time and balance is non-linear but not exponential.

### Step 2: Apply Regression
We will use the **Ending Balance** as the dependent variable and **Year** as the independent variable. The model can be fitted using a regression tool to find the best coefficients.

- **Exponential Decay Model**: 
  \[
  B(t) = B_0 e^{-kt}
  \]
  where $B(t)$ is the loan balance at time $t$, $B_0$ is the initial loan balance ($200,000), and $k$ is a decay constant to be determined.

- **Polynomial Regression Model**:
  A 2nd-degree polynomial might look like:
  \[
  B(t) = at^2 + bt + c
  \]
  where $a$, $b$, and $c$ are constants determined through regression.

### Step 3: Use Regression Software
I will now perform the regression analysis using the data provided.

It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can explain the process of performing regression manually or help with any other math-related questions in the meantime. Would you like to go over the details of regression models or another aspect of this problem?

Here are five follow-up questions for further exploration:
1. How would an exponential decay model compare to a linear regression for this data?
2. What does the decay constant represent in an exponential model, and how would it affect the loan balance over time?
3. How do different loan terms (years) and interest rates impact the loan balance schedule?
4. What is the benefit of using polynomial regression over exponential decay for financial data?
5. How could we use regression to predict the loan balance at any given year?

Tip: When choosing a regression model, always consider the shape of the data. Exponential decay works well for rapidly decreasing values, while polynomial fits handle more complex curves.

Kace does not have a very good credit rating. In January of 2021, he found a lending institution that was willing to loan him $200,000 for 15 years at a higher than normal interest rate of 6% because of his poor credit rating. Examine the table of decreasing loan balances over the 15-year period. Use regression to determine a curve of best fit for this data.

Learn how to apply regression analysis to financial data, using both exponential decay and polynomial regression models. Examine how a $200,000 loan with a 6% interest rate decreases over 15 years and determine the best fit for this data.

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