a. Give the design matrix, the observation vector, and the unknown parameter vector. Choose the correct design matrix X below.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:**

The given data produced by certain experiment is .Also given form of the function is

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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Question

a. Give the design matrix, the observation vector, and the unknown parameter vector. Choose the correct design matrix X below.

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:**

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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