ISYE 6414 - MIDTERM 1 PREP QUESTIONS ANSWERED CORRECTLY LATEST UPDATE 2026

ISYE 6414 - MIDTERM 1 PREP QUESTIONS ANSWERED CORRECTLY LATEST UPDATE 2026 If λ=1 - Answers we do not transform non-deterministic - Answers Regression analysis is one of the simplest ways we have in statistics to investigate the relationship between two or more variables in a ___ way random - Answers The response variable is a ___ variable, because it varies with changes in the predicting variable, or with other changes in the environment fixed - Answers The predicting variable is a ___ variable. It is set fixed, before the response is measured. simple linear regression - Answers 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 - Answers 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 - Answers 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 - Answers 1) Prediction 2) Modeling 3) Testing hypothesis Prediction - Answers 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 - Answers modeling the relationship between the response variable and the explanatory variables, or predicting variables Testing hypotheses - Answers of association relationships useful representation of reality - Answers We do not believe that the linear model represents a true representation of reality. Rather, we think that, perhaps, it provides a ___ β0 - Answers intercept parameter (the value at which the line intersects the y-axis) β1 - Answers slope parameter (slope of the line we are trying to fit) epsilon (ε) - Answers is the deviance of the data from the linear model to find β0 and β1 - Answers to find the line that describes a linear relationship, such that we fit this model. simple linear regression data structure - Answers 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: - Answers 1) identifying data structure 2) clearly stating the model assumptions linear regression assumptions - Answers 1) linearity 2) constant variance assumption 3) independence assumption linearity assumption - Answers 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 - Answers 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 - Answers 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 - Answers 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 - Answers 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. - Answers How can we get estimates of the regression coefficients or parameters in linear regression analysis? fitted values - Answers to be the regression line where the parameters are replaced by the estimated values of the parameters. Residuals - Answers are simply the difference between observed response and fitted values, and they are proxies of the error terms in the regression model MSE - Answers 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) - Answers 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 - Answers proxies for the deviances or the error terms sample variance estimator (s^2) - Answers the estimator of the variance of the error terms (is chi-square with n - 1 degrees of freedom) positive value for ß1 - Answers a direct relationship between the predicting variable x and the response variable y negative value of ß1 - Answers an inverse relationship between x and y. ß1 is close to zero. - Answers there is not a significant association between the predicting variable x, and the response variable y. ß1 hat - Answers is the estimated expected change in the response variable associated with one unit of change in the predicting variable. ß0 hat - Answers is the estimated expected value of the response variable, when the predicting variable equals zero we use ß1 hat - Answers 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 - Answers we always have to mention the statistical significance, whether statistically significantly positive, statistically significantly negative, or no statistical significance. estimated standard deviation - Answers in model summary, look for Residual standard error estimate of the variance (from output) - Answers we need to take the square of the residual standard error extrapolation - Answers not within the range of the observed axis, predicting larger than the values observed expectation of a linear combination of random variables - Answers is equal to the linear combination of the expectations. expectation of the estimator for the slope parameter is - Answers exactly ß1. ( ß1 ( ß1 hat is an unbiased estimator for ß1) unbiasedness - Answers the fact that the expectation of the estimator is exactly the true parameter that we're estimating sampling distribution of ß1 hat - Answers is a T distribution with N - 2 degrees of freedom. to obtain a confidence interval with (1-alpha)% confidence level - Answers we can center the confidence interval at the estimated value for β1, plus or minus the (1-alpha critical point T. T value - Answers the estimated value for ß1 /the standard error of the estimator. If that T value is large - Answers reject the null hypothesis that ß1 is equal to zero. If the null hypothesis is rejected, we interpret this that ß1 is statistically significant. Statistical significance means - Answers that ß1 is statistically different from zero. If the T value is larger than its critical point in absolute value, - Answers we say that the slope coefficient is statistically significantly different from c. If the P value is small, for example, smaller than .01, - Answers we would reject the new hypothesis. For ß1 greater than zero - Answers we're interested on the right tail of the distribution of the ß1 hat. For ß1 smaller than zero - Answers we're interested on the left tail. sample distribution of ß0 - Answers is also T distribution To perform statistical inference, we need to find: - Answers ● the estimated coefficient β1 and its variance along with the sample distribution of β1. ● the estimated coefficient for the intercept β0 and its variance along with the sample distribution ● whether the coefficient β1 is statistically significant. ● whether β1 is statistically positive. the estimated variance (of ß1) - Answers we need to take the square of the standard error function pt() - Answers stands for the probability of a t-distribution, gives left tail evaluating the quantile equal to the t-value. In order to get the right tail of that distribution we'd have to take one minus that probability. P-value - Answers is a measure of how rejectable the null hypothesis is. The smaller the p-value is, the more rejectable the null hypothesis is for the observed data, given the observed data. It's not the probability of rejecting the null hypothesis, nor is it the probability that the null hypothesis is true. x* under estimation - Answers it's really an average across all possible settings when we could observe x*. x* under prediction - Answers x* is considered an observation under a new setting, we focus on one particular setting. normal - Answers Y-hat has a ___ distribution. Thus, regression line is an unbiased estimator just like the estimators of the regression coefficients. x* equals - Answers x hat. variance is going to be smaller at the center of the average and is going to increase as we go away from the average as the predicted value x* is away from the average. - Answers The uncertainty in the estimated regression line is going to be higher confidence band - Answers If we take several values of x* and we construct such confidence intervals, we get ___ prediction contains two sources of uncertainty: - Answers 1. The new observation (because we are predicting the response under a new setting) 2. the parameter estimation (of the response for x*) Prediction interval is wider than then the confidence interval for the mean response. - Answers This is because we have additional uncertainty due to predicting under a new setting whereas the confidence intervals under estimation are reflecting an average across all settings for that specific value If the scatter plot of the residuals is not random around zero line - Answers relationship between x and y may not be linear, or the variances of the error terms are not equal, and the response data or the error terms are not independent residuals are clustered in two separated clusters - Answers means that the residuals may be correlated due to some clustering effect independence - Answers residual analysis cannot be used to check for the ___ assumption For checking normality, - Answers we can use the quantile plot, or normal probability plot If some of the assumptions do not hold - Answers then we interpret that the model fit is inadequate, but it does not mean that the regression is not useful. to model the nonlinear relationship - Answers we can transform X by some nonlinear function such as f(x) = x^a or f(x) = log(x) If λ=0 - Answers we actually use the normal logarithmic transformation If λ=-1 - Answers use the inverse of y, this is called the Box-Cox Transformation. outliers, - Answers are data points far from the majority of the data in x and/or y