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ISYE 6414 - Midterm 1 Prep Questions And Answers With Verified Updates
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σ^2 (sample distribution of the variance estimator) - is chi-squared distribution with n - 2 degrees of freedom (We lose two degrees of freedom because we replaced the two parameters ß0 and ß1 with their estimators to obtain the residuals.) constant variance assumption - which means that the...

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  • September 1, 2024
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ISYE 6414 - Midterm 1 Prep
σ^2 (sample distribution of the variance estimator) - is chi-squared distribution with n - 2 degrees
of freedom (We

lose two degrees of freedom because we replaced the two parameters ß0 and ß1 with

their estimators to obtain the residuals.)




constant variance assumption - which means that the variance (σ^2) of the error terms or
deviances is constant for the given population. A violation of this assumption means that the estimates
are not as efficient as they could be in estimating the true parameters



random - The response variable is a ___ variable, because it varies with changes in the predicting
variable, or with other changes in the environment



fixed - The predicting variable is a ___ variable. It is set fixed, before the response is measured.



simple linear regression - regression analysis involving one independent variable and one
dependent variable in which the relationship between the variables is approximated by a straight line



Multiple Linear Regression - A statistical method used to model the relationship between one
dependent (or response) variable and two or more independent (or explanatory) variables by fitting a
linear equation to observed data



polynomial regression - a regression model which does not assume a linear relationship; a
curvilinear correlation coefficient is computed (we can think of X and X-squared as two different
predicting variables)



three objectives in regression - 1) Prediction

2) Modeling

3) Testing hypothesis

,Prediction - We want to see how the response variable behaves in different settings. For example,
for a different location, if we think about a geographic prediction, or in time, if we think about temporal
prediction



Modeling - modeling the relationship between the response variable and the explanatory
variables, or predicting variables



Testing hypotheses - of association relationships



useful representation of reality - We do not believe that the linear model represents a true
representation of reality. Rather, we think that, perhaps, it provides a ___



β0 - intercept parameter (the value at which the line intersects the y-axis)



β1 - slope parameter (slope of the line we are trying to fit)



epsilon (ε) - is the deviance of the data from the linear model



to find β0 and β1 - to find the line that describes a linear relationship, such that we fit this model.



simple linear regression data structure - pairs of data consisting of a value for the response
variable,and a value for the predicting variable. And we have n such pairs



modeling framework for the simple linear regression: - 1) identifying data structure

2) clearly stating the model assumptions



linear regression assumptions - 1) linearity

2) constant variance assumption

3) independence assumption

, linearity assumption - mean zero assumption, means that the expected value of the errors is zero.

A violation of this assumption will lead to difficulties in estimating β0, and means that your model does
not include a necessary systematic component.



Independence Assumption - which means that the deviances are independent random variables.

Violation of this assumption can lead to misleading assessments of the strength of the regression.



If λ=1 - we do not transform



non-deterministic - Regression analysis is one of the simplest ways we have in statistics to
investigate the relationship between two or more variables in a ___ way



normality assumption - errors (ε) are normally distributed. This is needed for statistical inference,
for example, confidence or prediction intervals, and hypothesis testing. If this assumption is violated,
hypothesis tests and confidence and prediction intervals can be misleading.v



third parameter - the variance of the error terms (σ^2)



One approach is to minimize the sum of squared residuals or errors with respect to β0 and β1. This
translated into finding the line such that the total squared deviances from the line is minimum. -
How can we get estimates of the regression coefficients or parameters in linear

regression analysis?



fitted values - to be the regression line where the parameters are replaced

by the estimated values of the parameters.



Residuals - are simply the difference

between observed response and fitted values, and they are proxies of the error terms in

the regression model

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