Certainly! Let's go through the steps required to find the exponential regression equation and estimate the [tex]\( x \)[/tex] value for a given [tex]\( y \)[/tex] value. Here's the detailed solution:
### 1. Gather the Data Points
We are given the following data points:
[tex]\[
(-4.7, 10.7), (-7.8, 20.6), (-10.5, 30.2), (-15.6, 41), (-20.8, 56.1), (-22, 65.1)
\][/tex]
### 2. Transform the [tex]\( y \)[/tex]-Values
We will apply a logarithmic transformation to the [tex]\( y \)[/tex]-values to linearize the exponential relationship.
[tex]\[
y = a \cdot e^{bx} \Rightarrow \ln(y) = \ln(a) + bx
\][/tex]
### 3. Perform Linear Regression on Transformed Data
Using the transformed data [tex]\((x, \ln(y))\)[/tex], we perform linear regression to find the coefficients [tex]\( \ln(a) \)[/tex] and [tex]\( b \)[/tex]. Linear regression will give us a linear equation of the form:
[tex]\[
\ln(y) = bx + \ln(a)
\][/tex]
From this, we can determine [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
a = e^{\ln(a)}
\][/tex]
### 4. Round the Coefficients
The coefficients [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are rounded to two decimal places:
[tex]\[
a = 9.03 \quad \text{and} \quad b = -0.09
\][/tex]
Therefore, the estimated exponential regression equation becomes:
[tex]\[
y = 9.03 \cdot e^{-0.09x}
\][/tex]
### 5. Estimate [tex]\( x \)[/tex] for a Given [tex]\( y \)[/tex]-Value
To estimate the [tex]\( x \)[/tex] value when [tex]\( y \)[/tex] is 5.2, we substitute [tex]\( y = 5.2 \)[/tex] into the regression equation and solve for [tex]\( x \)[/tex]:
[tex]\[
5.2 = 9.03 \cdot e^{-0.09x}
\][/tex]
First, isolate the exponential term:
[tex]\[
\frac{5.2}{9.03} = e^{-0.09x}
\][/tex]
Then, take the natural logarithm of both sides:
[tex]\[
\ln\left(\frac{5.2}{9.03}\right) = -0.09x
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\ln\left(\frac{5.2}{9.03}\right)}{-0.09}
\][/tex]
After computing, we get:
[tex]\[
x \approx 6.0
\][/tex]
Rounding to one decimal place, the estimated [tex]\( x \)[/tex] value is:
[tex]\[
x = 6.0
\][/tex]
### Conclusion
Thus, the exponential regression equation is:
[tex]\[
y = 9.03 \cdot e^{-0.09x}
\][/tex]
When the [tex]\( y \)[/tex] value is 5.2, the estimated [tex]\( x \)[/tex] value is:
[tex]\[
x \approx 6.0
\][/tex]
