Exam 13: Multiple Regression Analysis

Rocket Experiments Narrative An engineer was investigating the relationship between the thrust of an experimental rocket (y), the percent composition of a secret chemical in the fuel (x₁), and the internal temperature of a chamber of the rocket (x₂). The engineer starts by fitting a quadratic model, but he believes that the full quadratic model is too complex and can be reduced by including only the linear terms and the interaction term. -Refer to Chemical Analysis Narrative. Use the statistical software output below to test whether the reduced model is adequate at the 0.05 level of significance. Complete Model Regression Analysis The regression equation is Analysis of Variance Reduced Model Regression Analysis The regression equation is Analysis of Variance

To check out whether the regressions assumption involving normality of the error terms (residuals) is valid, it is appropriate to construct a normal probability plot. If this plot forms a straight line from the lower-left-hand corner to the upper-right-hand corner, the error terms may be assumed to be normally distributed.

In a multiple regression model where four independent variables are included in the model, the percentage of explained variation in the dependent variable will be equal to the square root of the sum of the largest correlations between the dependent variable and the four independent variables.

Multiple regression analysis is a type of regression analysis in which several independent variables are used to estimate the value of an unknown dependent variable; hence, each of these predictor variables explains part of the total variation of the dependent variable.

Multicollinearity does not affect the F-test of the analysis of variance.

When the independent variables are correlated with one another in a multiple regression analysis, what is this condition called?

When an additional explanatory variable is introduced into a multiple regression model, the coefficient of multiple determination adjusted for degrees of freedom can never decrease.

Suppose a regression analysis based on the model with 15 observations produced SSE = 3.55, = 13.131, and = 125.1. In this case, what is the proportion of the total variability in y that is accounted for by and ?

In a multiple regression model, it is assumed that the residuals are normally distributed.

College Textbook Sales Narrative A publisher of college textbooks conducted a study to relate profit per text y to cost of sales x over a six-year period when its sales force (and sales costs) were growing rapidly. These inflation-adjusted data (in thousands of dollars) were collected: Expecting profit per book to rise and then plateau, the publisher fitted the model to the data. -Refer to College Textbook Sales Narrative. Use the values of SSR and Total SS in the printout to calculate Compare this value with the value given in the printout.

Demographic Variables and TV Narrative A statistician wanted to determine if the demographic variables of age, education, and income influence the number of hours of television watched per week. A random sample of 25 adults was selected to estimate the multiple regression model: , where y is the number of hours of television watched last week, is the age (in years), is the number of years of education, and is income (in $1000s). The computer output is shown below. The regression equation is S = 4.51 R-Sq = 34.8% Analysis of Variance -Refer to Demographic Variables and TV Narrative. Test the overall validity of the model at the 5% significance level.

A multiple regression equation includes five independent variables, and the coefficient of determination is 0.81. What is the percentage of the variation in y that is explained by the regression equation?

Discuss briefly what is meant by multicollinearity.

In a multiple regression analysis, the regression equation is obtained. The variable is quantitative variable, and the variable is a dummy variable with values 0 and 1. Given this information, we can interpret the slope coefficient (-3) on variable as follows: Holding constant, if the value of is changed from 0 to 1, the average value of y will decrease by 3 units.

Multicollinearity is present when there is a high degree of correlation between the dependent variable and all the independent variables included in the model.

In multiple regression, the prediction equation is the line that minimizes SSE, the sum of squares of the deviations of the observed values y from the predicted values .

Plots of the residuals against or against the individual independent variables often indicate departures from the assumptions required for an analysis of variance, and they also may suggest changes in the underlying model.

A multiple regression model has the form . As increases by one unit, holding constant, the value of y will increase by 9 units.

A three-variable multiple regression establishes an estimated multiple regression equation. Which of the following is a property of that equation?

In testing the validity of a multiple regression model, a large value of the F-test statistic is indicative of which of the following situations?