1. Show that the following relationship on the simple linear regression class notebook is true: E, x}-nx
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.
![**Problem Statement:**
**1.** Show that the following relationship in the simple linear regression class notebook is true:
\[
\frac{\sum_{i=1}^{n} x_i y_i - n \overline{X} \overline{Y}}{\sum_{i=1}^{n} x_i^2 - n \overline{X}^2} = \frac{\sum_{i=1}^{n} (x_i - \overline{X})(y_i - \overline{Y})}{\sum_{i=1}^{n} (x_i - \overline{X})^2}
\]
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**Explanation:**
The equation provided is a statement concerning a relationship in linear regression analysis. This identity shows an equivalence between two different formulaic expressions for calculating the slope coefficient (b) in a simple linear regression.
- The left side of the equation uses the formula that involves the sums of products of \(x_i\) and \(y_i\), alongside their means.
- The right side of the equation uses the covariance of \(x_i\) and \(y_i\), demonstrating that it’s equal to the ratio of covariance and variance.
This relationship is fundamental in deriving the least squares estimates in statistics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f8edec1-8d43-45b2-a95b-be2335431e6b%2F3b34f148-b5aa-4126-b742-f982a7d5ce82%2F8it6e4l_processed.png&w=3840&q=75)
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