x	1	2	3	4	5	6
y	473	398	307	214	170	129

Use exponential regression to find an exponential equation that best fits this data. Round both a and b to 4 decimal places.

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.

Use exponential regression to find an exponential equation that best fits this data. Round both a and b to 4 decimal places.
x = [1, 2, 3, 4, 5, 6]
y = [473, 398, 307, 214, 170, 129]

Find the exponential regression equation that best fits the data set (x = [1, 2, 3, 4, 5, 6], y = [473, 398, 307, 214, 170, 129]) using logarithmic transformation and linear regression. This solution covers advanced concepts of exponential modeling and is ideal for students in Grades 11-12.

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