**Exponential Regression Guide****Objective:**Use regression to determine an exponential function that best fits the provided data.**Data Table:**| X | 0 | 1 | 2 | 4 | 6 | 8 | 9 | 10 | 13 | 19 ||----|------|------|------|------|------|------|------|------|------|------|| Y | 21.98| 20.37| 19.18| 18.67| 18.28| 17.29| 15.82| 12.05| 11.23| 9.46 |**Traditional Form:** Typically, exponential regression is represented as:\[ y = a(b)^x \]**Alternative Form:** Sometimes, using different expressions for exponential regression can be advantageous:\[ y = e^{cx + d} \]**Task:** Find the exponential function in the form $ y = e^{cx + d} $, where:- $ c = \text{Number} $- $ d = \text{Number} $(Round your answers to three decimal places.)**Hints for Solution:**You can start with $ y = a(b)^x $ and manipulate it to find $ c $ and $ d $.Transform: \[ y = e^{cx + d} = e^d \cdot e^{cx} = (e^d)(e^c)^x = a(b)^x \]This implies: - $ a = e^d $- $ b = e^c $Use this relationship to solve for $ c $ and $ d $.

Given: Use the exponential function in the form y =ecx+d To Find: the value of c and d. Formula used: The regression coefficients formula for the exponential model of the form y =abx is given by the formula ln b=n∑x lny-∑x∑ln yn∑x2-∑x2 ln a=∑x2∑ln y-∑x∑x ln yn∑x2-∑x2 The exponential function of the form y =ecx+d can be written as y=ecxed Take a=ed, b=ec. Then it becomes y=a bx⋯⋯1. Let's calculate the following terms. ∑xlog(y) = 0 * log (21.98) + 1*log(20.37) + 2*log(19.18) + 4*log(18.67) + 6*log(18.28) + 8*log(17.29) + 9*log(15.82) + 10 * log(12.05) + 13 * log(11.23) + 19 * log(9.46) = 0 + (1.31) + 2*(1.28) + 4*(1.27) + 6*(1.26) + 8*(1.23) + 9*(1.2) + 10*(1.08) + 13*(1.05) + 19*(0.97) = 80.14 ∑log(y) =1.34 + (1.31) + (1.28) + (1.27) + (1.26) + (1.23) + (1.2) + (1.08) + (1.05) + (0.97) = 11.99 From the relation between ln and log10 , we have lnx=2.303 log10x. So, we have ∑xln(y)=2.303×80.14 =184.562 and ∑ln(y) =2.303×11.99=27.613 ∑(x)2 = 0 + 1 + 4 + 16 + 36 + 64 + 81 + 100 + 169 + 361 = 832 ∑x = 0 + 1 + 2 + 4 + 6 + 8 + 9 + 10 + 13 + 19 = 72 Therefore, by using the formula ln b=n∑x lny-∑x∑ln yn∑x2-∑x2, where n=10 we have the following ln(b) = [10*(184.562) - (72)* (27.613)][10*(832) - 72*72]ln(b) = -142.5163136ln(b)= - 0.045b=e- 0.045b=0.956 and ln(a) = [832*(27.613) -72* (184.562)][10*(832) - 72*72]ln(a) = 9685.5523136ln(a)= 3.0885a=e3.0885a=21.944 Since we want to write the equation in the form, y = ecx+ d So, we have the relation a=ed, b=ec. d=lnad=ln21.944 d=3.088 and c=lnbc=ln0.956c=-0.0449c≃-0.045 The values are approximated every time so the final value might be a little different from the exact answer. It's the approximate value of c and d. Approximated value (rounded to 3 decimal places) : c=-0.045d=3.088

Answered: Use regression to find an exponential…

Math

Algebra

Use regression to find an exponential function that best fits the data given below. ΧΟ 12 4 6 8 1 9 10 13 19 Y21.98 20.37 19.18 18.67 18.28 17.29 15.82 12.05 11.239.46

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

**Exponential Regression Guide**

**Objective:**
Use regression to determine an exponential function that best fits the provided data.

**Data Table:**

| X | 0 | 1 | 2 | 4 | 6 | 8 | 9 | 10 | 13 | 19 |
|----|------|------|------|------|------|------|------|------|------|------|
| Y | 21.98| 20.37| 19.18| 18.67| 18.28| 17.29| 15.82| 12.05| 11.23| 9.46 |

**Traditional Form:**
Typically, exponential regression is represented as:
\[ y = a(b)^x \]

**Alternative Form:**
Sometimes, using different expressions for exponential regression can be advantageous:
\[ y = e^{cx + d} \]

**Task:**
Find the exponential function in the form $ y = e^{cx + d} $, where:
- $ c = \text{Number} $
- $ d = \text{Number} $

(Round your answers to three decimal places.)

**Hints for Solution:**
You can start with $ y = a(b)^x $ and manipulate it to find $ c $ and $ d $.

Transform:
\[ y = e^{cx + d} = e^d \cdot e^{cx} = (e^d)(e^c)^x = a(b)^x \]

This implies:
- $ a = e^d $
- $ b = e^c $

Use this relationship to solve for $ c $ and $ d $.

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