Exam 12: Multiple Regression and Model Building

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: $s _ { b _ { 1 } } = 800 , s _ { b _ { 2 } } = 1000 , s _ { b _ { 3 } } = 1850$ -Assume today is Saturday morning and the weather forecast indicates sunny,excellent weather conditions for the rest of the day and that the overall model is useful in predicting the game attendance.Later today,there is a home baseball game for this team.If the current power rating of the team is 92,use the model given above and predict the attendance for today's game.

Below is a partial multiple regression computer output. Source SS df Model 32,774 5 Error 21,886 292 Total 54,660 297 Coefficient Variable Estimate Standard Error Intercept 22.02 -.18 .04 -.25 .12 -4.69 1.70 3.67 0.40 22.32 3.60 -Write the least squares prediction equation.

The general form of the quadratic multiple regression models is:

Consider the multiple regression model $y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \cdots + \beta _ { k } x _ { k } + \varepsilon$ )When using this model,we assume that at any given combination of values of x₁,x₂,... ,x_k the population of potential error term values has a ________ distribution.

The assumption of independent error terms in regression analysis is often violated when using time series data.

Below is a partial multiple regression ANOVA table. Source SS df Model 0.242 2 Error 0.105 3 -Test the overall usefulness of the model at alpha =.05.Calculate the F statistic and make your decision.

Below is a partial multiple regression computer output. Source SS df Model 32,774 5 Error 21,886 292 Total 54,660 297 -Determine the number of observations in the sample,the number of independent variables in the model,and the mean squared error.

A multiple regression model with 3 independent variables and 16 observations produced the following results: SSE = 15 and R²= 2/3.Complete the analysis of variance table and calculate the F statistic.

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 F statistic?

In comparing regression models,the regression model with the largest R² will also have the smallest standard error (s).

R²is defined as:

An acceptable residual plot exhibits:

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 -Test the overall usefulness of the model at alpha =.01.Calculate F and make your decision about whether the model is useful for prediction purposes.

For combinations of data within the experimental region,the least squares plane is the estimate of the plane of means.

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 explained variation?

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 explained variation?

If it is desired to include marital status in a multiple regression model by using the categories: single,married,separated,divorced,widowed,what will be the effect on the model?

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 -Calculate the proportion of the variation explained by the multiple 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 R²?

If we are predicting y when the values of the independent variables are x₀₁,x₀₂,…. ,x_0k,the farther the values of x₀₁,x₀₂,…. ,x_0k are from the center of the experimental region,the smaller the distance value and the more precise the associated confidence and prediction intervals.