To write an exponential regression equation for the given data, we need to find an equation of the form:

$y = a \cdot b^x$

Where:

• $a$ is the initial value (the value of $y$ when $x = 0$),
• $b$ is the growth rate.

We can find the values of $a$ and $b$ using exponential regression techniques, which typically involve a logarithmic transformation.

Steps to find the coefficients:

1. Transform the equation: Take the natural logarithm of both sides: $\ln(y) = \ln(a) + x \ln(b)$ This is now a linear equation in terms of $\ln(y)$ and $x$, which can be solved using linear regression.

2. Use linear regression: Fit the equation $\ln(y) = \ln(a) + x \ln(b)$ to the given data points.

Let's compute the values of $a$ and $b$.

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Data Set:

$x$	$y$
1	774
4	1032
7	1246
8	1471
9	1619

Now, I will perform the linear regression for $\ln(y)$ vs. $x$ to determine the coefficients $\ln(a)$ and $\ln(b)$.

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I'll calculate the coefficients now. numpy as np from scipy import stats

Given data

x_values = np.array([1, 4, 7, 8, 9]) y_values = np.array([774, 1032, 1246, 1471, 1619])

Apply natural log transformation to y-values

ln_y = np.log(y_values)

Perform linear regression on ln(y) vs x

slope, intercept, r_value, p_value, std_err = stats.linregress(x_values, ln_y)

Calculate a and b

a = np.exp(intercept) b = np.exp(slope)

Round a and b to the nearest thousandth

a_rounded = round(a, 3) b_rounded = round(b, 3)

a_rounded, b_roundedThe exponential regression equation for the given data, rounded to the nearest thousandth, is:

$y = 707.431 \cdot (1.094)^x$

Would you like a more detailed explanation of the process or have any other questions? Here are some related questions you might find helpful:

1. How can I interpret the coefficients $a$ and $b$ in an exponential model?
2. How do I perform exponential regression using software like Excel or Python?
3. What is the difference between exponential and linear regression?
4. How can I use this exponential model to predict values of $y$ for new $x$-values?
5. What happens if the data follows a different type of growth, like logistic growth, and how does that affect the model?

Tip: When fitting exponential models, always check the residuals to ensure the model fits the data well and the error is minimized.