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Exam 13: Multiple Regression Analysis

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Question 61

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A coefficient of multiple correlation is denoted by and equals the proportion of the total variation in the values of the dependent variable, y, that is explained by the estimated multiple regression of y on , , and possibly additional independent variables ( and so on).

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Question 62

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The stepwise regression analysis is best used as a preliminary tool for identifying which of a large number of variables should be considered in the model.

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Question 63

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Which of the following methods is used to help assess whether the regression model meets the assumption of having normally distributed residuals?

In a multiple regression analysis involving 50 observations and 5 independent variables, SST = 475 and SSE = 71.25. Then, the multiple coefficient of determination is 0.85.

To test the validity of a multiple regression model involving three predictor variables, which of the following is the best formulation of the null hypothesis to be tested?

If you wish to test the usefulness of a multiple regression model with three independent variables, the appropriate null and alternative hypotheses are vs.

Electric Usage Narrative The power company claims the amount of electricity used by a house (y) depends on = square metres of heated space, = mean outside temperature, and = mean hours of sunlight per day. Partial statistical software output is given below. Regression Analysis The regression equation is = 357 + 0.808 - 16.6 + 40 S = 267.7 R-sq = 82.0% R-sq(adj) = 76.6% Analysis of Variance -Refer to Electric Usage Narrative. Carry out three separate tests with a significance level of 0.05 to decide if , , and are significant.

Stepwise regression analysis is a procedure that is implemented by computer and is available in most statistical packages. It is used mainly to determine which of a large number of independent variables should be included in the model.

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. Interpret the coefficient .

The multiple coefficient of determination measures the proportion or percentage of variation in the dependent variable that is explained by the independent variables included in the model.

Stepwise regression is an iterative procedure that does which of the following?

In order to test the usefulness of a multiple regression model involving 5 predictor variables and 25 observations, the numerator and denominator degrees of freedom for the critical value of F are 4 and 24, respectively.

In a multiple regression model, the coefficient of determination (sometimes called multiple ) can be computed by simply squaring the largest correlation coefficient between the dependent variable and any independent variable.

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 Rocket Experiments Narrative. The engineer obtained a random sample of 66 measurements and computed the SSE for both the complete model and the reduced model. The values were 1477.8 and 1678.8, respectively. Perform the appropriate test of hypothesis to determine whether the reduced model is adequate for the engineer's use. Use = 0.05.

The computer output for a multiple regression analysis including 15 observations and three predictor variables, and , provides the following information: , , , , SE , SE , and SE a. Which, if any, of the independent variables and contribute information for the prediction of y? b. Give the least-squares prediction equation. c. What is the practical interpretation of the parameter

Life Expectancy Narrative An actuary wanted to develop a model to predict how long individuals will live. After consulting a number of physicians, she collected the age at death (y), the average number of hours of exercise per week ( ), the cholesterol level ( ), and the number of points that the individual's blood pressure exceeded the recommended value ( ). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below. The regression equation is S = 9.47 R-Sq = 22.5% Analysis of Variance -Refer to Life Expectancy Narrative. Interpret the coefficient .

Air Pollution Monitors Narrative An experiment was designed to compare several different types of air pollution monitors. Each monitor was set up and then exposed to different concentrations of ozone, ranging between 15 and 230 parts per million (ppm), for periods of 8-72 hours. Filters on the monitor were then analyzed, and the response of the monitor was measured. The results for one type of monitor showed a linear pattern. The results for another type of monitor are listed in the table. -Refer to Air Pollution Monitors Narrative. Does the model contribute significant information for the prediction of the monitor's response based on ozone exposure? Use the appropriate p-value to make your decision.

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. What conclusions can you draw from the accompanying residual plots?

Multiple linear regression is an extension of simple linear regression to allow for more than one dependent variable.

A three-variable multiple regression plane is positioned so as to minimize the sum of squared errors. This sum is given by which of the following expressions?