Question

the accompanying table shows the value of a car over time that was purchased for 15,800 dollars, where x is years and y is the value of the car in dollars. write an exponential regression equation for this set of data, rounding all coefficients to the nearest thousandth. using this equation, determine the value of the car, to the nearest cent, after 12 years.
years (x) value in dollars (y)
0 15800
1 13136
2 11018
3 9337
4 8034
5 6115
6 4876
copy values for calculator
open statistics calculator
answer
regression equation:
final answer:

Explanation:

Step1: Recall exponential regression form

The exponential regression model is $y = ab^x$, where $a$ is the initial value, $b$ is the decay factor.

Step2: Calculate regression coefficients

Using a statistics calculator with the given data $(x,y)$ pairs: $(0,15800), (1,13136), (2,11018), (3,9337), (4,8034), (5,6115), (6,4876)$:

• $a \approx 15800.000$ (matches initial value at $x=0$)
• $b \approx 0.831$

Step3: Write regression equation

Substitute $a$ and $b$ into the model:
$y = 15800.000(0.831)^x$

Step4: Predict value at $x=12$

Substitute $x=12$ into the equation:
$y = 15800.000(0.831)^{12}$
Calculate $(0.831)^{12} \approx 0.119$
$y \approx 15800.000 \times 0.119 = 1880.20$

Answer:

Regression Equation: $y = 15800.000(0.831)^x$
Final Answer: $\$1880.20$