ISYE 6414 - Midterm 1 Prep Questions And Answers With Verified Solutions

If λ=1 - Answer-we do not transform non-deterministic - Answer-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 - Answer-The response variable is a ___ variable, because it varies with changes in the predicting variable, or with other changes in the environment fixed - Answer-The predicting variable is a ___ variable. It is set fixed, before the response is measured. simple linear regression - Answer-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 - Answer-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 - Answer-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 - Answer-1) Prediction 2) Modeling 3) Testing hypothesis Prediction - Answer-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 predictionModeling - Answer-modeling the relationship between the response variable and the explanatory variables, or predicting variables Testing hypotheses - Answer-of association relationships useful representation of reality - Answer-We do not believe that the linear model represents a true representation of reality. Rather, we think that, perhaps, it provides a ___ β0 - Answer-intercept parameter (the value at which the line intersects the y-axis) β1 - Answer-slope parameter (slope of the line we are trying to fit) epsilon (ε) - Answer-is the deviance of the data from the linear model to find β0 and β1 - Answer-to find the line that describes a linear relationship, such that we fit this model. simple linear regression data structure - Answer-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: - Answer-1) identifying data structure 2) clearly stating the model assumptions linear regression assumptions - Answer-1) linearity 2) constant variance assumption 3) independence assumption linearity assumption - Answer-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 ant variance assumption - Answer-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 - Answer-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 - Answer-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 - Answer-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. - AnswerHow can we get estimates of the regression coefficients or parameters in linear regression analysis? fitted values - Answer-to be the regression line where the parameters are replaced by the estimated values of the parameters. Residuals - Answer-are simply the difference between observed response and fitted values, and they are proxies of the error terms in the regression model MSE - Answer-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) - Answer-is chi-squared distribution with n - 2 degrees of freedom (Welose two degrees of freedom because we replaced the two parameters ß0 and ß1 with their estimators to obtain the residuals.) epsilon i hat - Answer-proxies for the deviances or the error terms sample variance estimator (s^2) - Answer-the estimator of the variance of the error terms (is chi-square with n - 1 degrees of freedom) positive value for ß1 - Answer-a direct relationship between the predicting variable x and the response variable y negative value of ß1 - Answer-an inverse relationship between x and y. ß1 is close to zero. - Answer-there is not a significant association between the predicting variable x, and the response variable y. ß1 hat - Answer-is the estimated expected change in the response variable associated with one unit of change in the predicting variable. ß0 hat - Answer-is the estimated expected value of the response variable, when the predicting variable equals zero we use ß1 hat - Answer-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 - Answer-we always have to mention the statistical significance, whether statistically significantly positive, statistically significantly negative, or no statistical ated standard deviation - Answer-in model summary, look for Residual standard error estimate of the variance (from output) - Answer-we need to take the square of the residual standard error extrapolation - Answer-not within the range of the observed axis, predicting larger than the values observed expectation of a linear combination of random variables - Answer-is equal to the linear combination of the expectations. expectation of the estimator for the slope parameter is - Answer-exactly ß1. ( ß1 ( ß1 hat is an unbiased estimator for ß1) unbiasedness - Answer-the fact that the expectation of the estimator is exactly the true parameter that we're estimating sampling distribution of ß1 hat - Answer-is a T distribution with N - 2 degrees of freedom. to obtain a confidence interval with (1-alpha)% confidence level - Answer-we can center the confidence interval at the estimated value for β1, plus or minus the (1-alpha critical point T. T value - Answer-the estimated value for ß1 /the standard error of the estimator. If that T value is large - Answer-reject the null hypothesis that ß1 is equal to zero. If the null hypothesis is rejected, we interpret this that ß1 is statistically significant