Exam 12: Multiple Regression and Model Building

Given the regression model y = β₀ + β₁ X₁ + β₂ X₂ + β₃ X₁² + β₄X₂² + ε,if we wish to test the significance of higher order terms, (X₁² and X₂²)we would use the following test:

The standard error decreases if and only if the adjusted multiple coefficient of determination decreases.

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.With 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 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 = 5.465 R-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 Based on the multiple regression model given above,the point estimate of the monthly light fixture sales corresponding to second sample data is 49.82 or 49,820 units.This point estimate is calculated based on the assumption that the company spends $4000 on advertising,the price of the fixture is $60,and the fixture is being sold at 3 retail stores.Additional information related to this point estimate is given below. Obs. Advert Sales Fit StDev Fit Residual St Resid. 2 40.0 42.00 49.82 3.53 -7.82 -1.87 -Determine the 95% confidence interval for this point estimate and interpret its meaning.

Below is a partial multiple regression ANOVA table. Source SS df Model 0.242 2 Error 0.105 3 -What is the total sum of squares (total variation),explained variation,mean square error and the number of observations in the sample?

Consider the following partial computer output for a multiple regression model. Predictor Coefficient Standard Dev Constant 99.3883 X1 -0.007207 0.0031 X2 0.0011336 0.00122 X3 0.9324 0.373 Analysis of Variance SS Source df 31.308 Regression 3 9.378 -Consider the following analysis of variance table from a multiple regression model.Test the model for overall usefulness at

Below is a partial multiple regression ANOVA table. Source SS df 535.9569 1 1,167.5634 1 18.9886 1 Error 3,459.6803 8 -What is the total sum of squares (total variation)?

The management of a professional baseball team is in the process of determining the budget for next year.A major component of future revenue is attendance at the home games.In order to predict attendance at home games,the team's statistician has used a multiple regression model with dummy variables.The model is of the form: y = β₀+ β₁x₁+ β₂x₂+ β₃x₃+ ε where: Y = attendance at a home game x1= current power rating of the team on a scale from 0 to 100 before the game. x2and x3are dummy variables,and they are defined below. x2= 1,if weekend x2= 0,otherwise x3= 1,if weather is favourable x3= 0,otherwise After collecting the data based on 30 games from last year and implementing the above stated multiple regression model,the team statistician obtained the following least squares multiple regression equation: $\hat { y } = - 1050 + 250 x _ { 1 } + 2200 x _ { 2 } + 5400 x _ { 3 }$ The multiple regression compute output also indicated the following: $s _ { b _ { 1 } } = 800 , s _ { b _ { 2 } } = 1000 , s _ { b _ { 3 } } = 1850$ -Interpret the estimated model coefficient b1

Completely randomized analysis of variance models (one-way ANOVA)can always be converted to a multiple regression models with dummy independent variables.

R²is defined as the proportion of the observed variation in the dependent variable that is explained by the fitted regression model.

Multicollinearity between independent variables is serious when the correlation between pair(s)of dependent variables is _____.

Given the regression model y = β₀+ β₁X₁+ β₂X₁²+ ε,if we wish to test the significance of X₁²,the appropriate null hypothesis is:

Inclusion of redundant independent variables in the regression model increases the problem of __________.

In a regression equation,beta values are previously known parameter values.

In the quadratic regression model y = ?₀+ ?₁x + ?₂x²+ ?,if the value of the term ?₁increases,then the parabola shifts to the _________.

Below is a partial multiple regression computer output. Source SS df Model 32,774 5 Error 21,886 292 Total 54,660 297 -Test the overall usefulness of the model at

Write the least squares prediction equation.

Determine the 95% interval for β₂ and interpret its meaning

Below is a partially completed multiple regression analysis of variance (ANOVA)table.What is the value of R²? Source SS df Model 32,774 5 Error 21,886 292

Test the usefulness of variable x₂ in the model at

Below is a partial multiple regression computer output. Source SS df Model 32,774 5 Error 21,886 292 Total 54,660 297 -What is the number of independent variables in the model?