Exam 17: Multiple Regression

Some of the requirements for the error variable in a multiple regression model are that the standard deviation is a(n) ____________________ and the errors are ____________________.

In a multiple regression model, the value of the coefficient of determination has to fall between

Real Estate Builder: A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below. SUMMARY OUTPUT ANOVA -At the 0.01 level of significance, what conclusion should the builder draw regarding the inclusion of income in the regression model?

For the multiple regression model: , if x₂ were to increase by 5, holding x₁ and x₃ constant, the value of y will:

There are several clues to the presence of multicollinearity. One clue is when an independent variable is added or deleted, the regression coefficients for the other variables ____________________.

Life Expectancy 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 (x₁), the cholesterol level (x₂), and the number of points that the individual's blood pressure exceeded the recommended value (x₃). A random sample of 40 individuals was selected. The computer output of the multiple regression model is shown below. THE REGRESSION EQUATION IS y = 55.8 + 1.79x₁ -0.021x₂ - 0.061x₃ ANALYSIS OF VARIANCE -{Life Expectancy Narrative} Interpret the coefficient b₁.

When an explanatory variable is dropped from a multiple regression model, the adjusted coefficient of determination can increase.

Student's Final Grade: A statistics professor investigated some of the factors that affect an individual student's final grade in her course. She proposed the multiple regression model , where y is the final grade (out of 100 points), x₁ is the number of lectures skipped, x₂ is the number of late assignments, and x₃ is the midterm exam score (out of 100). The professor recorded the data for 50 randomly selected students. The computer output is shown below. THE REGRESSION EQUATION IS ANALYSIS OF VARIANCE -Does this data provide enough evidence to conclude at the 5% significance level that the model is useful in predicting the final grade?

The adjusted coefficient of determination is adjusted for the:

Suppose a multiple regression analysis involving 25 data points has and SSE = 36. Then, the number of the independent variables must be:

A multiple regression is called "multiple" because it has several explanatory variables.

Real Estate Builder: A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below. SUMMARY OUTPUT ANOVA -What percentage of the variability in house size is explained by this model?

In calculating the standard error of the estimate, , there are (n - k - 1) degrees of freedom, where n is the sample size and k is the number of independent variables in the model.

Real Estate Builder: A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below. SUMMARY OUTPUT ANOVA -At the 0.01 level of significance, what conclusion should the builder draw regarding the inclusion of education in the regression model?

Real Estate Builder: A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below. SUMMARY OUTPUT ANOVA -What are the numerator and denominator degrees of freedom for the F-statistic?

In a multiple regression model, the probability distribution of the error variable $\varepsilon$ is assumed to be:

Plot the residuals against the predicted values .

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

Real Estate Builder: A real estate builder wishes to determine how house size is influenced by family income, family size, and education of the head of household. House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is measured in years. A partial computer output is shown below. SUMMARY OUTPUT ANOVA -What is the value of the calculated F-test statistic that is missing from the output for testing whether the whole regression model is significant?

Consider the following statistics of a multiple regression model: n = 25, k = 5, b₁ = -6.31, and s _{$\varepsilon$} = 2.98. Can we conclude at the 1% significance level that x₁ and y are linearly related?