Exam 12: Multiple Regression and Model Building

Based on the multiple regression model given above,estimate the mathematics test score and calculate the value of the residual,if the percentage of teachers with a mathematics degree is 50.0,the average age is 43 and the average salary is $48,300 (48.3).The actual mathematics test score for these factors is 68.50.

A multiple linear regression analysis involving 45 observations resulted in the following least squares prediction equation: $\hat { y } = .408 + 1.3387 x _ { 1 } + 2.1 x _ { 2 }$ . The SSE for the above model is 49. Addition of two other independent variables to the model,resulted in the following multiple linear regression equation: $\hat { y } = 1.2 + 3 x _ { 1 } + 12 x _ { 2 } + 4 x _ { 3 } + 8 x _ { 4 }$ . The latter model's SSE is 40. -Determine the degrees of freedom regression (explained variation),degrees of freedom error (unexplained variation),and total degrees of freedom for the model with two independent variables.

Consider the following partial computer output for a multiple regression model. Predictor Coefficient Standard Dev Constant 99.3883 1 -0.007207 0.0031 2 0.0011336 0.00122 3 0.9324 0.373 Analysis of Variance Source 31.308 Regression 3 9.378 Error (residual) 16 -What is the value of R²?

A researcher in human resources has expressed concern about the differences in job satisfaction results across units within an organization. The researcher conducts a study to investigate what factors could account for the differences. The researcher looked at a random sample of units across the organization and used the factors of percentage of employees with a university degree, the average age of the employees, and the average salary of employees within a unit. The results of the study are presented below: Predictor Coef SE Coef Constant 35.178 7.595 Degree 0.22073 0.07131 Age 0.3353 0.1901 Salary 0.0930 0.1675 $s = 7.62090$ Analysis of Variance Source DF SS Regression 3 1053.09 Residual Error 32 1858.50 Source DF Seq SS Degree 1 672.10 Age 1 363.09 Salary 1 17.90 -Using the results above,what is the number of observations in the sample?

To test whether an individual independent variable makes a significant contribution to a multiple regression model,we would use a hypothesis test based on the ___ distribution.

Dummy variables take on the values of ______ or _____ and are used to model the effects of different levels of qualitative variables.

An application of the multiple regression model generated the following results involving the F test of the overall regression model: p - value = .0012,R² = .67 and s = .076.Thus,the null hypothesis,which states that none of the independent variables are significantly related to the dependent variable,should be rejected at the .01 level of significance.

Consider the following partial computer output for a multiple regression model. Predictor Coefficient Standard Deviation Constant 41.225 6.380 1.081 1.353 -18.404 4.547 Analysis of Variance Source Regression 2 2270.11 Error (residual) 26 3585.75 -What is the mean square error?

The ________ test is a statistical test used for testing a subset of one or more of the independent variables for significance.

A member of the provincial legislature has expressed concern about the differences in the mathematics test scores of grade 9 high school students across the province.She asks her research assistant to conduct a study to investigate what factors could account for the differences.The research assistant looked at a random sample of school districts across the province and used the factors of percentage of mathematics teachers in each district with a degree in mathematics,the average age of mathematics teachers,and the average salary of mathematics teachers Predictor Coef SE Coef Constant 35.178 7.595 Math Dgr 0.22073 0.07131 Age 0.3353 0.1901 Salary 0.0930 0.1675 s = 7.62090 Analysis of Variance Source DF SS Regression 3 1053.09 Residual Error 32 1858.50 Source DF Seq SS Math Dgr 1 672.10 Age 1 363.09 Salary 1 17.90 -What is the total sum of squares (total variation)?

In a regression model,at any given combination of values of the independent variables,the population of potential error terms is assumed to have an F-distribution.

Plotting the residuals will reveal possible violations of the __________ of error terms assumption.

A _______ plot is a residual plot that is used for the purpose of checking the normality assumption of the error terms in a multiple regression model.

A(n)_____ observation is an observation that causes the least squares point estimates to be substantially different from what they would be if the observation were removed from the data.

When the F test is used to test the overall significance of a multiple regression model,if the null hypothesis is rejected,it can be concluded that all of the independent variables x₁,x₂,…x_k are significantly related to the dependent variable y.

Test the usefulness of the variable X² in the model at

The manufacturer of a light fixture believes that the dollars spent on advertising,the price of the fixture,and the number of retail stores selling the fixture in a particular month influence the light fixture sales.The manufacturer randomly selects 10 months and collects the following data: Sales Advertising Price \# of stores 41 20 40 1 42 40 60 3 59 40 20 4 60 50 80 5 81 50 10 6 80 60 40 6 100 70 20 7 82 70 60 8 101 80 30 9 110 90 40 10 The sales are in thousands of units per month,the advertising is given in hundreds of dollars per month,the price is the unit retail price for the particular month.Using this data,the following computer output is obtained. The regression equation is Sales = 31.0 + 0.820 Advertising - 0.325 Price + 1.84 Stores Predictor Coef StDev T P Constant 30.992 7.728 4.01 0.007 Advertising 0.8202 0.5023 1.63 0.154 Price -0.32502 0.08935 -3.64 0.011 Stores 1.841 3.855 0.48 0.650 S = 5.465R-Sq = 96.7%R-Sq(adj)= 95.0% Analysis of Variance Source DF SS MS F P Regression 3 5179.2 1726.4 57.81 0.000 Residual Error 6 179.2 29.9 Total 9 5358.4 -Calculate the 95% prediction interval for this point estimate.

The variance inflation factor measures the relationship between the dependent variable and the rest of the independent variables in the regression model.

Below is a partial multiple regression computer output based on a quadratic regression model. Source SS df Model 29.44 2 Error 59.96 15 Standard Error Variable Coefficient Intercept 8.01 1.45 -1.35 0.55 0.46 0.43 -What is the value of the mean squared error?

Determine the 95% interval for β₄ and interpret its meaning.