In a statistical study, it is found that variables x and y are correlated as follows. Find the least squares regression line in this model.
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.
![**Question 10: Statistical Study on Variables Correlation**
In a statistical study, it is found that variables \( x \) and \( y \) are correlated as follows. Find the least squares regression line in this model.
**Explanation:**
In order to determine the relationship between the variables \( x \) and \( y \), you are tasked with finding the least squares regression line. The least squares regression line is a straight line that best fits the data points on a scatter plot. This line is used to predict the value of the dependent variable (\( y \)) based on the value of the independent variable (\( x \)).
To find the least squares regression line, you follow these steps:
1. **Calculate the Slope (\( m \) ) and Y-intercept (\( b \) )**:
- **Slope (\( m \) )**: This represents the rate of change in \( y \) for a unit change in \( x \). It is calculated as:
\[
m = \frac{n(\sum{xy}) - (\sum{x})(\sum{y})}{n(\sum{x^2}) - (\sum{x})^2}
\]
- **Y-intercept (\( b \) )**: This represents the value of \( y \) when \( x \) is 0. It is calculated as:
\[
b = \frac{\sum{y} - m(\sum{x})}{n}
\]
Where \( n \) is the number of data points, \( \sum{xy} \) is the sum of the products of corresponding \( x \) and \( y \) values, \( \sum{x} \) is the sum of \( x \) values, \( \sum{y} \) is the sum of \( y \) values, and \( \sum{x^2} \) is the sum of the squares of \( x \) values.
2. **Form the Least Squares Regression Line Equation**:
- Once you have the slope \( m \) and the y-intercept \( b \), you can form the regression line equation as:
\[
y = mx + b
\]
To perform these calculations, you will typically need a set of data points for \( x \) and \( y \). Once you process these with the formulas above, you will derive](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2ea4237-947f-4fb7-b8d4-fb858663c4d1%2Fac469701-ee11-4dad-aece-c4362d71fcff%2Fkin1s0a.jpeg&w=3840&q=75)
Step-by-step procedure to find the regression line using Excel:
- In Excel sheet, enter x and y in different columns.
- In Data, select Data Analysis and Choose Regression.
- In Input Y Range, select y.
- In Input X Range, select x.
- Click Labels in First Row.
- Click OK.
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