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Q: Is the random error in a regression equation the predicted error?

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The mean sum of squares due to error: this is the sum of the squares of the differences between the observed values and the predicted values divided by the number of observations.

the residual is the difference between the observed Y and the estimated regression line(Y), while the error term is the difference between the observed Y and the true regression equation (the expected value of Y). Error term is theoretical concept that can never be observed, but the residual is a real-world value that is calculated for each observation every time a regression is run. The reidual can be thought of as an estimate of the error term, and e could have been denoted as ^e.

Bias is systematic error. Random error is not.

I've included links to both these terms. Definitions from these links are given below. Correlation and regression are frequently misunderstood terms. Correlation suggests or indicates that a linear relationship may exist between two random variables, but does not indicate whether X causes Yor Y causes X. In regression, we make the assumption that X as the independent variable can be related to Y, the dependent variable and that an equation of this relationship is useful. Definitions from Wikipedia: In probability theory and statistics, correlation (often measured as a correlation coefficient) indicates the strength and direction of a linear relationship between two random variables. In statistics, regression analysis refers to techniques for the modeling and analysis of numerical data consisting of values of a dependent variable (also called a response variable) and of one or more independent variables (also known as explanatory variables or predictors). The dependent variable in the regression equation is modeled as a function of the independent variables, corresponding parameters ("constants"), and an error term. The error term is treated as a random variable. It represents unexplained variation in the dependent variable. The parameters are estimated so as to give a "best fit" of the data. Most commonly the best fit is evaluated by using the least squares method, but other criteria have also been used.

how to reduce the problem of random error and systematic error while doing an experiment

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includes both positive and negative terms.

The total squared error between the predicted y values and the actual y values

Random error, measurement error, mis-specification of model (overspecification or underspecification), non-normality, plus many more.

Regression analysis is based on the assumption that the dependent variable is distributed according some function of the independent variables together with independent identically distributed random errors. If the error terms were not stochastic then some of the properties of the regression analysis are not valid.

The mean sum of squares due to error: this is the sum of the squares of the differences between the observed values and the predicted values divided by the number of observations.

the residual is the difference between the observed Y and the estimated regression line(Y), while the error term is the difference between the observed Y and the true regression equation (the expected value of Y). Error term is theoretical concept that can never be observed, but the residual is a real-world value that is calculated for each observation every time a regression is run. The reidual can be thought of as an estimate of the error term, and e could have been denoted as ^e.

Bias is systematic error. Random error is not.

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A Stochastic error term is a term that is added to a regression equation to introduce all of the variation in Y that cannot be explained by the included Xs. It is, in effect, a symbol of the econometrician's ignorance or inability to model all the movements of the dependent variable.

I've included links to both these terms. Definitions from these links are given below. Correlation and regression are frequently misunderstood terms. Correlation suggests or indicates that a linear relationship may exist between two random variables, but does not indicate whether X causes Yor Y causes X. In regression, we make the assumption that X as the independent variable can be related to Y, the dependent variable and that an equation of this relationship is useful. Definitions from Wikipedia: In probability theory and statistics, correlation (often measured as a correlation coefficient) indicates the strength and direction of a linear relationship between two random variables. In statistics, regression analysis refers to techniques for the modeling and analysis of numerical data consisting of values of a dependent variable (also called a response variable) and of one or more independent variables (also known as explanatory variables or predictors). The dependent variable in the regression equation is modeled as a function of the independent variables, corresponding parameters ("constants"), and an error term. The error term is treated as a random variable. It represents unexplained variation in the dependent variable. The parameters are estimated so as to give a "best fit" of the data. Most commonly the best fit is evaluated by using the least squares method, but other criteria have also been used.

If the regression sum of squares is the explained sum of squares. That is, the sum of squares generated by the regression line. Then you would want the regression sum of squares to be as big as possible since, then the regression line would explain the dispersion of the data well. Alternatively, use the R^2 ratio, which is the ratio of the explained sum of squares to the total sum of squares. (which ranges from 0 to 1) and hence a large number (0.9) would be preferred to (0.2).

When we use linear regression to predict values, we input a given x value and we use the equation of the correlation line to predict the y values. Sometimes we want to know how spread out the y values are. We look at the difference between the predicted and the actual y values. These differences are called residual and they are either positive if the y value is more than the estimated y value or negative if it is less. So for example if the observed value is 10 and the predicted one is 15, the residual is 15-10=5. Now we can find the residual for each y value in our data set and square it. Then we can take the average of those squares. Last, we take the square root of the average of the squared residuals and this is the RMS or root mean square error. The units are the same as the y values. If the RMS error is big, then the y values are not too close to the predicted ones on the y value and the our line does not provide as good of a model to predict values. If it is small, the y values are well predicted by the regression line. For a horizontal line, the RMS error is the same as the standard deviation. r is the regression coefficient and it measures how closely clustered the points are relative to the standard deviaton. The RMS error measures the spread in the original y units.

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