A certain experiment produces the data (1, 1.6), (2, 2.7), (3, 3.1), (4, 3.8), and (5, 3.9). Describe the model that produces a least-squares fit of these points by a function of the form y =B₁x + ₂x. Such a function might arise, for example, as the revenue from the sale of x units of a product, when the amount offered for sale affects the price to be set for the product. Answer parts (a) through (c) below.
A certain experiment produces the data (1, 1.6), (2, 2.7), (3, 3.1), (4, 3.8), and (5, 3.9). Describe the model that produces a least-squares fit of these points by a function of the form y =B₁x + ₂x. Such a function might arise, for example, as the revenue from the sale of x units of a product, when the amount offered for sale affects the price to be set for the product. Answer parts (a) through (c) below.
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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a. Give the design matrix, the observation
b. Find the associated least-squares curve for the data
![### Least-Squares Fit Model for Experimental Data
**Problem Description:**
A certain experiment produces the following data points:
- (1, 1.6)
- (2, 2.7)
- (3, 3.1)
- (4, 3.8)
- (5, 3.9)
You are asked to describe the model that produces a least-squares fit of these points by a function of the form \( y = \beta_1 x + \beta_2 x^2 \). Such a function might arise, for example, as the revenue from the sale of \( x \) units of a product when the amount offered for sale affects the price to be set for the product.
**Task:**
Answer parts (a) through (c) below.
---
**Explanation:**
The problem involves fitting a quadratic model to the given data points using the least-squares method. The general form of a quadratic model is:
\[ y = \beta_1 x + \beta_2 x^2 \]
Where:
- \( y \) is the dependent variable (e.g., revenue).
- \( x \) is the independent variable (e.g., units of product).
- \( \beta_1 \) and \( \beta_2 \) are the parameters of the model that need to be determined.
### Steps to Follow:
1. **Data Representation:**
- Organize the given data points in tabular form.
- Each data point has two values: \( x \) (independent variable) and \( y \) (dependent variable).
2. **Formulate the System of Equations:**
- Using the least-squares method, set up the equations that will help determine \( \beta_1 \) and \( \beta_2 \).
- This involves using summations and solving a system of linear equations.
3. **Solve for Parameters \( \beta_1 \) and \( \beta_2 \):**
- Apply algebraic methods or numerical techniques to solve the system and find the values of \( \beta_1 \) and \( \beta_2 \).
4. **Fit the Model:**
- Use the determined parameters to write the final model equation.
- Plot the fitted curve (quadratic) along with the original data points for visualization.
5. **Part (a) through (c) Responses:**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5246d2a-5796-4784-a543-e596d3b5542c%2Fd2560c0f-f7c9-48b6-8d10-d284802183e5%2Ff3oi6w8_processed.png&w=3840&q=75)
Transcribed Image Text:### Least-Squares Fit Model for Experimental Data
**Problem Description:**
A certain experiment produces the following data points:
- (1, 1.6)
- (2, 2.7)
- (3, 3.1)
- (4, 3.8)
- (5, 3.9)
You are asked to describe the model that produces a least-squares fit of these points by a function of the form \( y = \beta_1 x + \beta_2 x^2 \). Such a function might arise, for example, as the revenue from the sale of \( x \) units of a product when the amount offered for sale affects the price to be set for the product.
**Task:**
Answer parts (a) through (c) below.
---
**Explanation:**
The problem involves fitting a quadratic model to the given data points using the least-squares method. The general form of a quadratic model is:
\[ y = \beta_1 x + \beta_2 x^2 \]
Where:
- \( y \) is the dependent variable (e.g., revenue).
- \( x \) is the independent variable (e.g., units of product).
- \( \beta_1 \) and \( \beta_2 \) are the parameters of the model that need to be determined.
### Steps to Follow:
1. **Data Representation:**
- Organize the given data points in tabular form.
- Each data point has two values: \( x \) (independent variable) and \( y \) (dependent variable).
2. **Formulate the System of Equations:**
- Using the least-squares method, set up the equations that will help determine \( \beta_1 \) and \( \beta_2 \).
- This involves using summations and solving a system of linear equations.
3. **Solve for Parameters \( \beta_1 \) and \( \beta_2 \):**
- Apply algebraic methods or numerical techniques to solve the system and find the values of \( \beta_1 \) and \( \beta_2 \).
4. **Fit the Model:**
- Use the determined parameters to write the final model equation.
- Plot the fitted curve (quadratic) along with the original data points for visualization.
5. **Part (a) through (c) Responses:**
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