Tell about the meaning of Sxx and Sxy in simple linear regression, especially the meaning of those formulas

Question

Cheryl King · Accepted Answer

Sxx&amp;nbsp;- sum of the squares of the difference between each x and the mean x value and&amp;nbsp;Sxy&amp;nbsp;- sum of the product of the difference between x its means and the difference between y and its mean.&amp;nbsp;Sxx=∑(x−x―)(x−x―)&amp;nbsp;and&amp;nbsp;Sxy=∑(x−x―)(y−y―). When illustrated in textbooks, both of them are frequently rearranged into equivalent (different) forms.

psor32 · Accepted Answer

Sxx=∑x2−(∑x)2n=∑x2−nx―2Sxy&amp;nbsp;- the result when you replace one of the x&#039;s with a y.Sxy=∑xy−∑x∑yn=∑xy−nx―y―And the product-moment correlation coefficient:r=∑xy−nx―y―(∑x2−nx―2)(∑y2−ny―2)=SxySxxSyyor to find the gradient of the best-fit line&amp;nbsp;y=a+bx:y−y―=b(x−x―),where&amp;nbsp;b=SxyS×

Jeffrey Jordon · Accepted Answer

When we want to study the relationship between two variables, we can use simple linear regression to build a model that predicts the value of one variable based on the value of the other variable. To estimate the parameters of the regression model, we use the least squares method, which involves calculating the sums of squares

S_{x x}

and

S_{x y}

.

S_{x x}

represents the sum of squared deviations of the independent variable (

x

) from its mean (

\bar{x}

):

S_{x x} = \sum_{i = 1}^{n} (x_{i} - \bar{x})^{2}

S_{x y}

represents the sum of the product of the deviations of the independent variable (

x

) and the dependent variable (

y

) from their respective means (

\bar{x}

and

\bar{y}

):

S_{x y} = \sum_{i = 1}^{n} (x_{i} - \bar{x}) (y_{i} - \bar{y})

These quantities are used in the calculation of the slope and intercept of the regression line:

{\hat{β}}_{1} = \frac{S_{x y}}{S_{x x}}

{\hat{β}}_{0} = \bar{y} - {\hat{β}}_{1} \bar{x}

where

{\hat{β}}_{1}

is the estimated slope of the line, and

{\hat{β}}_{0}

is the estimated intercept.

S_{x x}

measures the variability of the independent variable, while

S_{x y}

measures the covariance between the independent variable and the dependent variable. These quantities are important in understanding the relationship between the two variables and in making predictions based on the regression model.
In simple terms,

S_{x x}

measures how spread out the independent variable is around its mean, while

S_{x y}

measures how much the independent and dependent variables vary together. The slope of the regression line (

{\hat{β}}_{1}

) tells us how strong the relationship is between the two variables, and the intercept (

{\hat{β}}_{0}

) represents the predicted value of the dependent variable when the independent variable is zero.
To summarize,

S_{x x}

and

S_{x y}

are crucial quantities in simple linear regression that help us estimate the parameters of the model and comprehend the relationship between the variables.

Vasquez · Accepted Answer

Solution:In simple linear regression, we are interested in finding the best-fitting line that describes the relationship between two variables X and Y. The line is described by an intercept β0 and a slope β1, such that:Y=β0+β1X+ϵ,where ϵ is the error term that represents the variability in Y that is not explained by the linear relationship with X.To estimate the values of β0 and β1, we need to use the sample data. The sample data consists of n pairs of observations (xi,yi), where i=1,2,…,n. We can use two key statistics, the sample covariance Sxy and the sample variance Sxx, to estimate β1 and β0.The sample variance Sxx is calculated as:Sxx=1n−1∑i=1n(xi−x¯)2,where x¯ is the sample mean of X. Sxx measures the variability of X around its mean.The sample covariance Sxy is calculated as:Sxy=1n−1∑i=1n(xi−x¯)(yi−y¯),where y¯ is the sample mean of Y. Sxy measures the linear association between X and Y.The slope of the line can be estimated as:β1^=SxySxx.This represents the change in Y associated with a one-unit change in X. The intercept can be estimated as:β0^=y¯−β1^x¯.Once we have estimated β0^ and β1^, we can use them to predict the value of Y for any given value of X. This is done by plugging the value of X into the equation of the line:y^=β0^+β1^x.In summary, the sample covariance Sxy measures the linear association between X and Y, while the sample variance Sxx measures the variability of X around its mean. These two statistics are used to estimate the slope and intercept of the line that best describes the relationship between X and Y.

RizerMix · Accepted Answer

Explanation:In statistics, simple linear regression is a method used to model the relationship between two variables, where one variable (the dependent variable) is assumed to be a linear function of the other variable (the independent variable). The method is widely used to make predictions and understand the nature of the relationship between the variables.To estimate the parameters of the regression model, we use the least squares method, which involves calculating the sums of squares Sxx and Sxy.Sxx represents the sum of squared deviations of the independent variable (x) from its mean (x¯):Sxx=∑i=1n(xi−x¯)2Sxy represents the sum of the product of the deviations of the independent variable (x) and the dependent variable (y) from their respective means (x¯ and y¯):Sxy=∑i=1n(xi−x¯)(yi−y¯)These quantities are used in the calculation of the slope and intercept of the regression line:β^1=SxySxxβ^0=y¯−β^1x¯where β^1 is the estimated slope of the line, and β^0 is the estimated intercept.Sxx represents the variability of the independent variable, while Sxy represents the covariance between the independent variable and the dependent variable. These quantities are important in understanding the relationship between the two variables and in making predictions based on the regression model.To put it simply, Sxx measures the spread of the independent variable around its mean, while Sxy measures the extent to which the independent and dependent variables vary together. The slope of the regression line (β^1) is a measure of the strength and direction of the relationship between the variables, and the intercept (β^0) represents the predicted value of the dependent variable when the independent variable is zero.In summary, Sxx and Sxy are important quantities in simple linear regression that help us estimate the parameters of the model and understand the relationship between the variables.

What is the meaning of 'Sxx' and 'Sxy' in simple

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