Exam 13: Multiple Regression Analysis

A multiple regression model involves 5 predictor variables and 30 observations. If we want to test the hypotheses at the 5% significance level, what will the critical value of the rejection region be?

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. Find s on the printout. Confirm that .

Personal Spending and Personal Income Is personal spending linearly related to orders for durable goods and personal income? A recent study reported the amounts of personal spending (in trillions of dollars), amount spent on durable goods (in billions of dollars), and personal income (in trillions of dollars). A statistical package was used to fit a linear regression model, producing the output below. R² = 95.9% R²(adj) = 95.0%, s = 0.0144 with 12 - 3 = 9 df. -Refer to Personal Spending and Personal Income. Use the computer output shown above to calculate 95% confidence intervals for the intercept and the partial regression coefficients.

In multiple regression, the descriptor "multiple" refers to more than one dependent variable.

In multiple regression analysis, when the response surface (the graphical depiction of the regression equation) hits every single point, the sum of squares for error SSE = 0, the standard error of estimate = 0, and the coefficient of determination = 1.

Personal Spending and Personal Income Is personal spending linearly related to orders for durable goods and personal income? A recent study reported the amounts of personal spending (in trillions of dollars), amount spent on durable goods (in billions of dollars), and personal income (in trillions of dollars). A statistical package was used to fit a linear regression model, producing the output below. R² = 95.9% R²(adj) = 95.0%, s = 0.0144 with 12 - 3 = 9 df. -Refer to Personal Spending and Personal Income. Based on your confidence intervals in the previous question, does the amount spent on durable goods have any predictive power beyond that provided by the other independent variables for determining personal spending? Explain.

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 .

In multiple regression analysis, the coefficient of determination is sometimes called multiple .

The coefficient of multiple determination is calculated by dividing the regression sum of squares by the total sum of squares (SSR/SST) and subtracting that value from 1.

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. Is there enough evidence at the 5% significance level to infer that the cholesterol level and the age at death are negatively linearly related? Justify your conclusion.

If we want to relate a random variable y to two independent variables and , a regression hyperplane is the three-dimensional equivalent of a regression line that minimizes the sum of the squared vertical deviations between the sample points suspended in y vs. vs. space and their associated multiple regression estimates, all of which lie on this hyperplane.

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. Is there sufficient evidence at the 1% significance level to indicate that hours of television watched and education are negatively linearly related? Justify your conclusion.

Chemical Comparisons Narrative A chemist was interested in examining the effects of three chemicals on a chemical process yield. Let , i = 1, 2, 3, represent the effects of the three chemicals, respectively, and y be the process yield. The following output was generated using statistical software. Regression Analysis S = 14.01 R-Sq = 95.1% R-Sq(adj) = 92.1 Analysis of Variance -Refer to Chemical Comparisons Narrative. Which variable, if any, should be removed from the model if a 0.05 level of significance is desired?

For which of the following quantities is the adjusted multiple coefficient of determination adjusted??

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. Do the data provide sufficient evidence to indicate that the model contributes information for the prediction of y? What is the p-value for this test, and what does it mean?

If a multiple regression model includes ten or more predictor variables, it is almost certain that changes in the predictor variables cause changes in the response variable y.

Personal Spending and Personal Income Is personal spending linearly related to orders for durable goods and personal income? A recent study reported the amounts of personal spending (in trillions of dollars), amount spent on durable goods (in billions of dollars), and personal income (in trillions of dollars). A statistical package was used to fit a linear regression model, producing the output below. R² = 95.9% R²(adj) = 95.0%, s = 0.0144 with 12 - 3 = 9 df. -Refer to Personal Spending and Personal Income. Predict the level of personal spending when amount spent on durable goods is 130 and personal income is at 5.10.

Personal Spending and Personal Income Is personal spending linearly related to orders for durable goods and personal income? A recent study reported the amounts of personal spending (in trillions of dollars), amount spent on durable goods (in billions of dollars), and personal income (in trillions of dollars). A statistical package was used to fit a linear regression model, producing the output below. R² = 95.9% R²(adj) = 95.0%, s = 0.0144 with 12 - 3 = 9 df. -Refer to Personal Spending and Personal Income. Use the computer output shown above to test the hypotheses at the 5% significance level. What is your conclusion?

Which of the following correctly describes the coefficient of multiple determination?

In stepwise regression procedure, what occurs if two independent variables are highly correlated?