least-squares regression line
Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
![### Analyzing a Data Set
A data set is provided for analysis with specific tasks outlined:
#### Given Data:
```
x: 1, 1, 4, 6, 6
y: 5.0, 6.3, 4.2, 3.4, 1.8
```
#### Tasks:
1. **Scatter Diagram and Relationship Analysis**
2. **Determining the Least-Squares Regression Line**
3. **Graphing the Least-Squares Regression Line**
---
#### Part (a) - Scatter Diagram and Relationship
*Choose the correct graph below:*
- **Option A** depicts the scatter plot where the following coordinate points (x, y) are plotted: (1, 5.0), (1, 6.3), (4, 4.2), (6, 3.4), and (6, 1.8).
There appears to be a **linear, negative** relationship between the variables x and y.
---
#### Part (b) - Least-Squares Regression Line
Given:
\[
\bar{x} = 3.6667, \quad s_x = 2.2509, \quad \bar{y} = 3.6833, \quad s_y = 1.8809, \quad r = -0.9511
\]
**Determine the least-squares regression line using the formula:**
\[
\hat{y} = b_0 + b_1x
\]
Where:
\[
b_1 = r \left( \frac{s_y}{s_x} \right)
\]
\[
b_0 = \bar{y} - b_1\bar{x}
\]
Calculate:
\[
b_1 = -0.9511 \left( \frac{1.8809}{2.2509} \right) \approx -0.795
\]
\[
b_0 = 3.6833 - (-0.795 \cdot 3.6667) \approx 6.588
\]
Thus, the equation of the least-squares regression line is:
\[
\hat{y} = 6.588 - 0.795x
\]
---
#### Part (c) - Graphing the Regression Line
*Choose the correct graph below:*
- **Option B** correctly represents](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F240ac775-9f02-4c90-873e-56631fdfab69%2Fcc689690-be58-4ce4-81f5-f352be95834b%2Fem0l8xm_processed.jpeg&w=3840&q=75)
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