To find the exponential regression equation in the form $y = a \cdot b^x$ that best fits the data, we will perform the following steps:

Data:

• $x = [1, 2, 3, 4, 5, 6]$
• $y = [473, 398, 307, 214, 170, 129]$

We can apply a logarithmic transformation to the data, fit it to a linear regression model, and then transform it back into the exponential form.

The transformed form is: $\ln(y) = \ln(a) + x \cdot \ln(b)$ We will find $a$ and $b$ using this linearized form, and then convert back to the original exponential model. Let me calculate that for you.The exponential regression equation that best fits the data is:

$y = 651.9199 \cdot (0.7643)^x$

Both $a$ and $b$ are rounded to four decimal places.

Would you like to see more details on the process, or do you have any other questions?

Here are 5 related questions to expand your understanding:

1. How do you apply logarithmic transformations to data?
2. What is the significance of the coefficients $a$ and $b$ in an exponential regression model?
3. How can you interpret the $R^2$ value from the regression analysis?
4. Can you use exponential regression for data that grows exponentially instead of decaying?
5. How would you handle outliers in an exponential regression analysis?

Tip: Exponential regression is useful for modeling data that decays or grows at a constant percentage rate per unit increase in the independent variable.