Find the linear curve fit coefficients. Use equations (4a) and (4b) to find these coefficients and determine the units of the coefficients, c1 and c2. Note that the units for these coefficients will not be the same in our two cases. Further note that the calibration coefficients are intermediate quantities, not final results. It is probably best to keep them to four significant digits to avoid round-off errors in subsequent calculations. Lastly, plot the line representing the curve fit. It should appear as a best fit, if you’ve done everything correctly. n is the number of data pairs in the curve-fit. the yi ’s arethe ordinate values (the less precisely known). the xi ’s are the abscissa values (the more precisely known).
Find the linear curve fit coefficients. Use equations (4a) and (4b) to find these coefficients and determine the units of the coefficients, c1 and c2. Note that the units for these coefficients will not be the same in our two cases. Further note that the calibration coefficients are intermediate quantities, not final results. It is probably best to keep them to four significant digits to avoid round-off errors in subsequent calculations. Lastly, plot the line representing the curve fit. It should appear as a best fit, if you’ve done everything correctly. n is the number of data pairs in the curve-fit. the yi ’s arethe ordinate values (the less precisely known). the xi ’s are the abscissa values (the more precisely known).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Find the linear curve fit coefficients. Use equations (4a) and (4b) to find these coefficients and determine the units of the coefficients, c1 and c2. Note that the units for these coefficients will not be the same in our two cases. Further note that the calibration coefficients are intermediate quantities, not final results. It is probably best to keep them to four significant digits to avoid round-off errors in subsequent calculations. Lastly, plot the line representing the curve fit. It should appear as a best fit, if you’ve done everything correctly. n is the number of data pairs in the curve-fit. the yi ’s arethe ordinate values (the less precisely known). the xi ’s are the abscissa values (the more precisely known).
Transcribed Image Text:### Pressure and Measurement Correlation Table
#### Date:
(leave this blank for the student to fill in the relevant date)
#### Group:
(leave this blank for the student to fill in the relevant group name)
This table shows the correlation between true pressure (`p_true`), analog pressure readings (`Analog`), and digital voltage readings (`Digital`).
| p_true (psi) | Analog (psi) | Digital (V) |
|--------------|--------------|-------------|
| 5 | 5 | 0.2872 |
| 10 | 10 | 0.3691 |
| 15 | 14 | 0.4497 |
| 35 | 34 | 0.775 |
| 55 | 54 | 1.107 |
| 75 | 74 | 1.4368 |
| 95 | 95 | 1.7653 |
| 105 | 104 | 1.9300 |
#### Explanation and Analysis:
- **p_true (psi)**: Indicates the true pressure in pounds per square inch (psi).
- **Analog (psi)**: Represents the analog pressure reading also in psi, which aims to approximate the true pressure.
- **Digital (V)**: Represents the digital reading in volts (V) corresponding to the pressure.
#### Observations:
1. As the `p_true` (psi) increases, both the `Analog` (psi) and `Digital` (V) values also increase.
2. The analog readings (`Analog`) closely match the true pressure (`p_true`), with minor deviations.
3. The digital readings (`Digital`) show a corresponding increase in voltage as the true pressure increases; these readings can be used to calibrate or verify the pressure using digital instruments.
This table is a practical reference for understanding the relationship between different measurement methods for pressure and could be beneficial for students studying instrumentation and measurement techniques in engineering or physical sciences.
#### Graphical Representation (Hypothetical Explanation):
1. **Analog vs True Pressure**: If we plot `Analog` (psi) on the y-axis and `p_true` (psi) on the x-axis, we would expect nearly a straight line with a slope close to 1, indicating a high correlation between the analog measurement and the true pressure.
2. **Digital Voltage vs True Pressure
![### Formulas for Linear Regression Coefficients
In the context of linear regression, the coefficients \( c_1 \) and \( c_2 \) can be calculated using the following formulas:
#### Formula for \( c_1 \)
\[
c_1 = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{n \sum x_i^2 - \left(\sum x_i\right)^2}
\]
This equation, marked as (4a), represents the slope of the regression line.
#### Formula for \( c_2 \)
\[
c_2 = \frac{\sum y_i \sum x_i^2 - \sum x_i y_i \sum x_i}{n \sum x_i^2 - (\sum x_i)^2}
\]
This equation, marked as (4b), represents the y-intercept of the regression line.
### Explanation
1. **Symbols and Notations**:
- \( n \) is the number of data points.
- \( x_i \) and \( y_i \) are the individual data points.
- \( \sum x_i \) represents the sum of all \( x_i \) values.
- \( \sum y_i \) represents the sum of all \( y_i \) values.
- \( \sum x_i y_i \) represents the sum of the product of \( x_i \) and \( y_i \) values.
- \( \sum x_i^2 \) represents the sum of the squares of \( x_i \) values.
#### Graphical Representation:
There are no graphical elements in this section.
### Application
These formulas are used to determine the best-fitting linear relationship between two variables in a dataset, which is crucial for predictive modeling and statistical analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e366a85-c23b-450a-9bc7-d2ea0be52d3f%2Fd288648b-ab5d-4668-b6dd-09a2c3398ebc%2Fvw5x04_processed.png&w=3840&q=75)
Transcribed Image Text:### Formulas for Linear Regression Coefficients
In the context of linear regression, the coefficients \( c_1 \) and \( c_2 \) can be calculated using the following formulas:
#### Formula for \( c_1 \)
\[
c_1 = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{n \sum x_i^2 - \left(\sum x_i\right)^2}
\]
This equation, marked as (4a), represents the slope of the regression line.
#### Formula for \( c_2 \)
\[
c_2 = \frac{\sum y_i \sum x_i^2 - \sum x_i y_i \sum x_i}{n \sum x_i^2 - (\sum x_i)^2}
\]
This equation, marked as (4b), represents the y-intercept of the regression line.
### Explanation
1. **Symbols and Notations**:
- \( n \) is the number of data points.
- \( x_i \) and \( y_i \) are the individual data points.
- \( \sum x_i \) represents the sum of all \( x_i \) values.
- \( \sum y_i \) represents the sum of all \( y_i \) values.
- \( \sum x_i y_i \) represents the sum of the product of \( x_i \) and \( y_i \) values.
- \( \sum x_i^2 \) represents the sum of the squares of \( x_i \) values.
#### Graphical Representation:
There are no graphical elements in this section.
### Application
These formulas are used to determine the best-fitting linear relationship between two variables in a dataset, which is crucial for predictive modeling and statistical analysis.
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