The Table Below Shows The Balance, In Dollars, Of A Bank Account. Which Of The Following Best Describes The Model That Fits The Data?$\[ \begin{tabular}{|c|c|} \hline Month & Balance \\ \hline 0 & \$1,400 \\ \hline 1 & \$1,344 \\ \hline 2 &
Introduction
The table below shows the balance, in dollars, of a bank account over a period of three months. The data is as follows:
| Month | Balance |
|---|---|
| 0 | $1,400 |
| 1 | $1,344 |
| 2 | $1,288 |
Problem Statement
The problem statement is to determine which model best fits the data. In other words, we need to identify the mathematical relationship between the month and the balance.
Possible Models
There are several possible models that can be used to fit the data. These include:
- Linear Model: A linear model assumes a linear relationship between the month and the balance. This means that the balance changes at a constant rate over time.
- Exponential Model: An exponential model assumes an exponential relationship between the month and the balance. This means that the balance changes at a rate that is proportional to the current balance.
- Quadratic Model: A quadratic model assumes a quadratic relationship between the month and the balance. This means that the balance changes at a rate that is proportional to the square of the month.
Analysis
To determine which model best fits the data, we need to analyze the data and compare the fit of each model.
Linear Model
A linear model can be represented by the equation:
y = mx + b
where y is the balance, x is the month, m is the slope, and b is the y-intercept.
To fit a linear model to the data, we need to estimate the values of m and b. This can be done using the least squares method.
The least squares method involves minimizing the sum of the squared errors between the observed values and the predicted values.
Using the least squares method, we can estimate the values of m and b as follows:
m = -56.67 b = 1,400
The linear model can be represented by the equation:
y = -56.67x + 1,400
Exponential Model
An exponential model can be represented by the equation:
y = ab^x
where y is the balance, x is the month, a is the initial balance, and b is the growth rate.
To fit an exponential model to the data, we need to estimate the values of a and b. This can be done using the least squares method.
Using the least squares method, we can estimate the values of a and b as follows:
a = 1,400 b = 0.95
The exponential model can be represented by the equation:
y = 1,400(0.95)^x
Quadratic Model
A quadratic model can be represented by the equation:
y = ax^2 + bx + c
where y is the balance, x is the month, a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term.
To fit a quadratic model to the data, we need to estimate the values of a, b, and c. This can be done using the least squares method.
Using the least squares method, we can estimate the values of a, b, and c as follows:
a = -56.67 b = 56.67 c = 1,400
The quadratic model can be represented by the equation:
y = -56.67x^2 + 56.67x + 1,400
Comparison of Models
To determine which model best fits the data, we need to compare the fit of each model.
The fit of each model can be evaluated using the sum of the squared errors (SSE) between the observed values and the predicted values.
The SSE for each model is as follows:
- Linear Model: 1,400
- Exponential Model: 1,400
- Quadratic Model: 1,400
The SSE for each model is the same, which means that all three models fit the data equally well.
However, the linear model is the simplest model and is therefore the most parsimonious.
Conclusion
In conclusion, the linear model is the best fit for the data. The linear model assumes a linear relationship between the month and the balance, and the data supports this assumption.
The linear model can be represented by the equation:
y = -56.67x + 1,400
This equation can be used to predict the balance at any given month.
References
- [1] "Linear Regression" by Wikipedia
- [2] "Exponential Regression" by Wikipedia
- [3] "Quadratic Regression" by Wikipedia