a = X y b= 1 2 3 4 978 1464 2445 4040 Use regression to find an exponential equation that best fits the data above. The equation has form y = ab* where: 5 6 6672 10601
a = X y b= 1 2 3 4 978 1464 2445 4040 Use regression to find an exponential equation that best fits the data above. The equation has form y = ab* where: 5 6 6672 10601
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Regression Analysis to Determine Exponential Growth**
The following data set consists of pairs of x and y values:
\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
y & 978 & 1464 & 2445 & 4040 & 6672 & 10601 \\
\hline
\end{array}
\]
### Problem Statement
Use regression to find an exponential equation that best fits the data above. The equation has the form \( y = ab^x \) where:
- \( a \) is (to be determined)
- \( b \) is (to be determined)
### Steps to Solve:
1. **Logarithm Transformation:**
- Take the natural logarithm (ln) of both sides to linearize the equation \( y = ab^x \).
- Thus, \( \ln(y) = \ln(a) + x \ln(b) \).
- Let \( Y = \ln(y) \), \( A = \ln(a) \), and \( B = \ln(b) \).
- The equation \( \ln(y) = \ln(a) + x \ln(b) \) converts to \( Y = A + Bx \), which is a linear equation.
2. **Linear Regression:**
- Perform linear regression on \( \ln(y) \) versus \( x \) to determine the values of \( A \) and \( B \).
3. **Exponentiation:**
- After finding \( A \) and \( B \) using linear regression, exponentiate \( A \) to find \( a \).
- Exponentiate \( B \) to find \( b \).
4. **Result:**
- The values of \( a \) and \( b \) give you the parameters for the exponential equation \( y = ab^x \).
Please complete the regression analysis using appropriate statistical tools or software to find \( a \) and \( b \).
**Note:** This guide is designed for educational purposes to help students understand the process of deriving an exponential equation from given data using regression analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F79384b4f-f6f5-4976-8d61-12e63a60eeb0%2F2ae53b2e-3d11-4b1e-99d5-292bf01782f2%2Fvaldlkr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Regression Analysis to Determine Exponential Growth**
The following data set consists of pairs of x and y values:
\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
y & 978 & 1464 & 2445 & 4040 & 6672 & 10601 \\
\hline
\end{array}
\]
### Problem Statement
Use regression to find an exponential equation that best fits the data above. The equation has the form \( y = ab^x \) where:
- \( a \) is (to be determined)
- \( b \) is (to be determined)
### Steps to Solve:
1. **Logarithm Transformation:**
- Take the natural logarithm (ln) of both sides to linearize the equation \( y = ab^x \).
- Thus, \( \ln(y) = \ln(a) + x \ln(b) \).
- Let \( Y = \ln(y) \), \( A = \ln(a) \), and \( B = \ln(b) \).
- The equation \( \ln(y) = \ln(a) + x \ln(b) \) converts to \( Y = A + Bx \), which is a linear equation.
2. **Linear Regression:**
- Perform linear regression on \( \ln(y) \) versus \( x \) to determine the values of \( A \) and \( B \).
3. **Exponentiation:**
- After finding \( A \) and \( B \) using linear regression, exponentiate \( A \) to find \( a \).
- Exponentiate \( B \) to find \( b \).
4. **Result:**
- The values of \( a \) and \( b \) give you the parameters for the exponential equation \( y = ab^x \).
Please complete the regression analysis using appropriate statistical tools or software to find \( a \) and \( b \).
**Note:** This guide is designed for educational purposes to help students understand the process of deriving an exponential equation from given data using regression analysis.
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