ISYE 6414 LATEST 2026 TEST PAPER QUESTIONS AND
SOLUTIONS RATED A+
✔✔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.
✔✔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
✔✔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.
✔✔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
✔✔MSE - ✔✔The estimator for sigma square is sigma square hat, and is the
sum of the squared residuals, divided by n - 2.
, ✔✔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.)
✔✔epsilon i hat - ✔✔proxies for the deviances or the error terms
✔✔sample variance estimator (s^2) - ✔✔the estimator of the variance of the error terms
(is chi-square with n - 1 degrees of freedom)
✔✔positive value for ß1 - ✔✔a direct relationship
between the predicting variable x and the response variable y
✔✔negative value of ß1 - ✔✔an inverse relationship between x and y.
✔✔ß1 is close to zero. - ✔✔there is not a significant association between the predicting
variable x, and the response variable y.
✔✔ß1 hat - ✔✔is the estimated expected change in the response variable associated
with
one unit of change in the predicting variable.
✔✔ß0 hat - ✔✔is the estimated expected value of the response variable, when the
predicting variable equals zero
✔✔we use ß1 hat - ✔✔when we interpret whether the relationship between x and y is
positive, negative, or
there is no relationship.
✔✔when we make statistical statements
about the relationship - ✔✔we always have to mention the statistical significance,
whether
statistically significantly positive, statistically significantly negative, or no statistical
significance.
✔✔estimated standard deviation - ✔✔in model summary, look for Residual standard
error
✔✔estimate of the variance (from output) - ✔✔we need to take the square of the
residual standard error
✔✔extrapolation - ✔✔not within the range of the observed axis, predicting larger than
the values observed
SOLUTIONS RATED A+
✔✔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.
✔✔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
✔✔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.
✔✔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
✔✔MSE - ✔✔The estimator for sigma square is sigma square hat, and is the
sum of the squared residuals, divided by n - 2.
, ✔✔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.)
✔✔epsilon i hat - ✔✔proxies for the deviances or the error terms
✔✔sample variance estimator (s^2) - ✔✔the estimator of the variance of the error terms
(is chi-square with n - 1 degrees of freedom)
✔✔positive value for ß1 - ✔✔a direct relationship
between the predicting variable x and the response variable y
✔✔negative value of ß1 - ✔✔an inverse relationship between x and y.
✔✔ß1 is close to zero. - ✔✔there is not a significant association between the predicting
variable x, and the response variable y.
✔✔ß1 hat - ✔✔is the estimated expected change in the response variable associated
with
one unit of change in the predicting variable.
✔✔ß0 hat - ✔✔is the estimated expected value of the response variable, when the
predicting variable equals zero
✔✔we use ß1 hat - ✔✔when we interpret whether the relationship between x and y is
positive, negative, or
there is no relationship.
✔✔when we make statistical statements
about the relationship - ✔✔we always have to mention the statistical significance,
whether
statistically significantly positive, statistically significantly negative, or no statistical
significance.
✔✔estimated standard deviation - ✔✔in model summary, look for Residual standard
error
✔✔estimate of the variance (from output) - ✔✔we need to take the square of the
residual standard error
✔✔extrapolation - ✔✔not within the range of the observed axis, predicting larger than
the values observed
