5 3 -1 2 135 y 0 1 9 5 3 9 6 4 a) Construct a scatter plot of these data b) Given that SST estimate of Bo and Bi 60, SSzy = 56.626, y = = 3.7143, x = 4.0, calculate the least squa
5 3 -1 2 135 y 0 1 9 5 3 9 6 4 a) Construct a scatter plot of these data b) Given that SST estimate of Bo and Bi 60, SSzy = 56.626, y = = 3.7143, x = 4.0, calculate the least squa
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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![### Example Data, Scatter Plot, and Least Squares Estimation
#### Sample Data
The dataset provided consists of paired values of \( x \) and \( y \) as shown below:
\[
\begin{array}{cc}
x & y \\
\hline
5 & 3 \\
3 & 5 \\
-1 & 0 \\
2 & 1 \\
9 & 9 \\
6 & 5 \\
4 & 3 \\
\end{array}
\]
#### Task a: Construct a Scatter Plot
To visualize the relationship between \( x \) and \( y \), we can create a scatter plot where each pair \((x, y)\) is represented as a point in a two-dimensional coordinate system.
**Explanation of the Scatter Plot:** Plot each \( x \) value on the horizontal axis and the corresponding \( y \) value on the vertical axis. For example, the point corresponding to the pair \((5, 3)\) will be plotted at the coordinate (5, 3).
#### Task b: Calculate the Least Squares Estimate
Given the following statistical measures:
- \( SS_{xx} = 60 \)
- \( SS_{xy} = 56.626 \)
- \( \bar{y} = 3.7143 \)
- \( \bar{x} = 4.0 \)
We can estimate the parameters \( \beta_0 \) and \( \beta_1 \) of the least squares regression line using the formulae:
- **Slope (\( \beta_1 \))**: \( \beta_1 = \frac{SS_{xy}}{SS_{xx}} \)
- **Intercept (\( \beta_0 \))**: \( \beta_0 = \bar{y} - \beta_1 \bar{x} \)
Plugging in the given values:
\[ \beta_1 = \frac{56.626}{60} \approx 0.9438 \]
\[ \beta_0 = 3.7143 - (0.9438 \times 4.0) \approx 3.7143 - 3.7752 \approx -0.0609 \]
Thus, the least squares estimates are:
- \( \beta_1 \approx 0.9438 \)
- \( \beta_0 \approx -0.0609 \)
These](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e08cb41-5b2e-4118-8784-a24243109431%2Fa9a84f7c-5296-4c29-a34e-a0e9c1a56f57%2Ftv6tt87_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Example Data, Scatter Plot, and Least Squares Estimation
#### Sample Data
The dataset provided consists of paired values of \( x \) and \( y \) as shown below:
\[
\begin{array}{cc}
x & y \\
\hline
5 & 3 \\
3 & 5 \\
-1 & 0 \\
2 & 1 \\
9 & 9 \\
6 & 5 \\
4 & 3 \\
\end{array}
\]
#### Task a: Construct a Scatter Plot
To visualize the relationship between \( x \) and \( y \), we can create a scatter plot where each pair \((x, y)\) is represented as a point in a two-dimensional coordinate system.
**Explanation of the Scatter Plot:** Plot each \( x \) value on the horizontal axis and the corresponding \( y \) value on the vertical axis. For example, the point corresponding to the pair \((5, 3)\) will be plotted at the coordinate (5, 3).
#### Task b: Calculate the Least Squares Estimate
Given the following statistical measures:
- \( SS_{xx} = 60 \)
- \( SS_{xy} = 56.626 \)
- \( \bar{y} = 3.7143 \)
- \( \bar{x} = 4.0 \)
We can estimate the parameters \( \beta_0 \) and \( \beta_1 \) of the least squares regression line using the formulae:
- **Slope (\( \beta_1 \))**: \( \beta_1 = \frac{SS_{xy}}{SS_{xx}} \)
- **Intercept (\( \beta_0 \))**: \( \beta_0 = \bar{y} - \beta_1 \bar{x} \)
Plugging in the given values:
\[ \beta_1 = \frac{56.626}{60} \approx 0.9438 \]
\[ \beta_0 = 3.7143 - (0.9438 \times 4.0) \approx 3.7143 - 3.7752 \approx -0.0609 \]
Thus, the least squares estimates are:
- \( \beta_1 \approx 0.9438 \)
- \( \beta_0 \approx -0.0609 \)
These
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