Question

The growth of a $40 000 investment is shown in the table.

Time (years) Value ($)
0 40 000
1 42 600
2 45 369
3 48 318
4 51 459
5 54 803

Use the exponential regression function on your calculator to get the function. Round a to the nearest dollar and b to the nearest thousandth.

Answer 1

Problem #6579. The growth of a $40 000 investment is shown in the table.

Time (years) Value ($) 0 40 000

1 42 600

2 45 369

3 48 318

4 51 459

5 54 803

Use the exponential regression function to get the function. Round $b$ to the nearest dollar and $a$ to the nearest thousandth.

Solution We are to find such $a$ and $b$, such that the following equality $V \approx b \cdot a^T$ is the most "precise". This equality is equivalent to $\log V \approx \log b + T \log a$ (we can think about this equality as about standard $L_2$ Gaussian regression $\log V = \log b + T \log a + \varepsilon$, where $\varepsilon$ are Gaussian errors). Using standard formulas for the estimation of intercept and slope: $\widehat{\log a} = \frac{cov(\log V, T)}{S^2(T)} = 0.03$ and $\widehat{\log b} = \overline{\log V} - \overline{T} \cdot \widehat{\log a} = 4.6$ (here $\widehat{A}$ denotes mean of the sample $A$, $cov(A, B)$ denotes the covariance of samples $A$ and $B$). Hence $a = 10^{0.03} \approx 1.0715$ and $b \approx 10^{4.6} \approx 39810.71$.

Answer $V = 39810.71 \cdot 1.0715^T$ or if we round $V = 39811 \cdot 1.072^T$.