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Exam 16: Regression Models for Nonlinear Relationships

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The coefficient of determination R2 cannot be used to compare the linear and quadratic models,because:

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Question 61

Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.For the estimated log-log model,interpret the estimated coefficient of ln(PepsiPrice). Using Excel's regression,the linear model PepsiSales = β0 + β1PepsiPrice + β2ColaPrice + ε and the log-log model ln(PepsiSales)= β0 + β1ln(PepsiPrice)+ β2ln(ColaPrice)+ ε have been estimated as follows: Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.For the estimated log-log model,interpret the estimated coefficient of ln(PepsiPrice). Refer to Exhibit 16.7.For the estimated log-log model,interpret the estimated coefficient of ln(PepsiPrice).

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Question 62

Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm For the assumed exponential model ln(Temp)= β<sub>0</sub> + β<sub>1</sub>Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.How many minutes must elapse after the brewing in order to cool the coffee to 158 <sup>o</sup>F? For the assumed exponential model ln(Temp)= β0 + β1Time + ε,the following Excel regression partial output is available. Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm For the assumed exponential model ln(Temp)= β<sub>0</sub> + β<sub>1</sub>Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.How many minutes must elapse after the brewing in order to cool the coffee to 158 <sup>o</sup>F? Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm For the assumed exponential model ln(Temp)= β<sub>0</sub> + β<sub>1</sub>Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.How many minutes must elapse after the brewing in order to cool the coffee to 158 <sup>o</sup>F? Refer to Exhibit 16-4.How many minutes must elapse after the brewing in order to cool the coffee to 158 oF?

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Question 63

Exhibit 16.2.Typically,the sales volume declines with an increase of a product price.It has been observed,however,that for some luxury goods the sales volume may increase when the price increases.The following Excel output illustrates this rather unusual relationship. Exhibit 16.2.Typically,the sales volume declines with an increase of a product price.It has been observed,however,that for some luxury goods the sales volume may increase when the price increases.The following Excel output illustrates this rather unusual relationship. Refer to Exhibit 16.2.For the considered range of the price,the relationship between Price and Sales should be described by a: Refer to Exhibit 16.2.For the considered range of the price,the relationship between Price and Sales should be described by a:

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Question 64

Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.For the estimated log-log model,interpret the estimated coefficient of ln(ColaPrice). Using Excel's regression,the linear model PepsiSales = β0 + β1PepsiPrice + β2ColaPrice + ε and the log-log model ln(PepsiSales)= β0 + β1ln(PepsiPrice)+ β2ln(ColaPrice)+ ε have been estimated as follows: Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.For the estimated log-log model,interpret the estimated coefficient of ln(ColaPrice). Refer to Exhibit 16.7.For the estimated log-log model,interpret the estimated coefficient of ln(ColaPrice).

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Question 65

Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. For the assumed cubic and log-log regression models,Demand = β<sub>0</sub> + β<sub>1</sub>Price + β<sub>2</sub>Price<sup>2</sup> + β<sub>3</sub>Price<sup>3</sup> + ε and ln(Demand)= β<sub>0</sub> + β<sub>1</sub>ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.Using the cubic model,what is the predicted demand when the price is $200? For the assumed cubic and log-log regression models,Demand = β0 + β1Price + β2Price2 + β3Price3 + ε and ln(Demand)= β0 + β1ln(Price)+ ε,the following regression results are available: Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. For the assumed cubic and log-log regression models,Demand = β<sub>0</sub> + β<sub>1</sub>Price + β<sub>2</sub>Price<sup>2</sup> + β<sub>3</sub>Price<sup>3</sup> + ε and ln(Demand)= β<sub>0</sub> + β<sub>1</sub>ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.Using the cubic model,what is the predicted demand when the price is $200? Refer to Exhibit 16.5.Using the cubic model,what is the predicted demand when the price is $200?

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Question 66

Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. For the assumed cubic and log-log regression models,Demand = β<sub>0</sub> + β<sub>1</sub>Price + β<sub>2</sub>Price<sup>2</sup> + β<sub>3</sub>Price<sup>3</sup> + ε and ln(Demand)= β<sub>0</sub> + β<sub>1</sub>ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.Assuming that the sample correlation coefficient between Demand and is 0.956,what is the predicted demand for a price of $250 found by the model with better fit? For the assumed cubic and log-log regression models,Demand = β0 + β1Price + β2Price2 + β3Price3 + ε and ln(Demand)= β0 + β1ln(Price)+ ε,the following regression results are available: Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. For the assumed cubic and log-log regression models,Demand = β<sub>0</sub> + β<sub>1</sub>Price + β<sub>2</sub>Price<sup>2</sup> + β<sub>3</sub>Price<sup>3</sup> + ε and ln(Demand)= β<sub>0</sub> + β<sub>1</sub>ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.Assuming that the sample correlation coefficient between Demand and is 0.956,what is the predicted demand for a price of $250 found by the model with better fit? Refer to Exhibit 16.5.Assuming that the sample correlation coefficient between Demand and Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. For the assumed cubic and log-log regression models,Demand = β<sub>0</sub> + β<sub>1</sub>Price + β<sub>2</sub>Price<sup>2</sup> + β<sub>3</sub>Price<sup>3</sup> + ε and ln(Demand)= β<sub>0</sub> + β<sub>1</sub>ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.Assuming that the sample correlation coefficient between Demand and is 0.956,what is the predicted demand for a price of $250 found by the model with better fit? is 0.956,what is the predicted demand for a price of $250 found by the model with better fit?

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Question 67

Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.What is the percentage of variations in the sales of Pepsi explained by the estimated linear model? Using Excel's regression,the linear model PepsiSales = β0 + β1PepsiPrice + β2ColaPrice + ε and the log-log model ln(PepsiSales)= β0 + β1ln(PepsiPrice)+ β2ln(ColaPrice)+ ε have been estimated as follows: Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.What is the percentage of variations in the sales of Pepsi explained by the estimated linear model? Refer to Exhibit 16.7.What is the percentage of variations in the sales of Pepsi explained by the estimated linear model?

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Question 68

Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.For the estimated linear model,when the price of Cola is held constant what is the predicted change in the Pepsi sales if the price of Pepsi increases by 10 cents? Using Excel's regression,the linear model PepsiSales = β0 + β1PepsiPrice + β2ColaPrice + ε and the log-log model ln(PepsiSales)= β0 + β1ln(PepsiPrice)+ β2ln(ColaPrice)+ ε have been estimated as follows: Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.For the estimated linear model,when the price of Cola is held constant what is the predicted change in the Pepsi sales if the price of Pepsi increases by 10 cents? Refer to Exhibit 16.7.For the estimated linear model,when the price of Cola is held constant what is the predicted change in the Pepsi sales if the price of Pepsi increases by 10 cents?

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Question 69

Exhibit 16-1.The following Excel scatterplot with the fitted quadratic regression equation illustrates the observed relationship between productivity and the number of hired workers. Exhibit 16-1.The following Excel scatterplot with the fitted quadratic regression equation illustrates the observed relationship between productivity and the number of hired workers. Refer to Exhibit 16.1.Assuming that the number of hired workers must be integer,how many workers should be hired in order to achieve the highest productivity? Refer to Exhibit 16.1.Assuming that the number of hired workers must be integer,how many workers should be hired in order to achieve the highest productivity?

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Question 70

Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.Discuss the choice between the linear model and the log-log model. Using Excel's regression,the linear model PepsiSales = β0 + β1PepsiPrice + β2ColaPrice + ε and the log-log model ln(PepsiSales)= β0 + β1ln(PepsiPrice)+ β2ln(ColaPrice)+ ε have been estimated as follows: Exhibit 16-7.It is believed that the sales volume of one liter Pepsi bottles depends on the price of the bottle and the price of one liter bottle of Coca Cola.The following data has been collected for a certain sales region. Using Excel's regression,the linear model PepsiSales = β<sub>0</sub> + β<sub>1</sub>PepsiPrice + β<sub>2</sub>ColaPrice + ε and the log-log model ln(PepsiSales)= β<sub>0</sub> + β<sub>1</sub>ln(PepsiPrice)+ β<sub>2</sub>ln(ColaPrice)+ ε have been estimated as follows: Refer to Exhibit 16.7.Discuss the choice between the linear model and the log-log model. Refer to Exhibit 16.7.Discuss the choice between the linear model and the log-log model.

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Question 71

Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. For the assumed cubic and log-log regression models,Demand = β<sub>0</sub> + β<sub>1</sub>Price + β<sub>2</sub>Price<sup>2</sup> + β<sub>3</sub>Price<sup>3</sup> + ε and ln(Demand)= β<sub>0</sub> + β<sub>1</sub>ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.What is the percentage of variations in ln(Demand)explained by the log-log regression equation? For the assumed cubic and log-log regression models,Demand = β0 + β1Price + β2Price2 + β3Price3 + ε and ln(Demand)= β0 + β1ln(Price)+ ε,the following regression results are available: Exhibit 16.5.The following data shows the demand for an airline ticket dependent on the price of this ticket. For the assumed cubic and log-log regression models,Demand = β<sub>0</sub> + β<sub>1</sub>Price + β<sub>2</sub>Price<sup>2</sup> + β<sub>3</sub>Price<sup>3</sup> + ε and ln(Demand)= β<sub>0</sub> + β<sub>1</sub>ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.What is the percentage of variations in ln(Demand)explained by the log-log regression equation? Refer to Exhibit 16.5.What is the percentage of variations in ln(Demand)explained by the log-log regression equation?

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Question 72

A quadratic regression model is a special type of a polynomial regression model.

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Question 73

For the quadratic equation For the quadratic equation ,which of the following expressions must be zero in order to minimize or maximize the predicted y? ,which of the following expressions must be zero in order to minimize or maximize the predicted y?

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Question 74

Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm For the assumed exponential model ln(Temp)= β<sub>0</sub> + β<sub>1</sub>Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the regression equation for making predictions concerning the coffee temperature? For the assumed exponential model ln(Temp)= β0 + β1Time + ε,the following Excel regression partial output is available. Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm For the assumed exponential model ln(Temp)= β<sub>0</sub> + β<sub>1</sub>Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the regression equation for making predictions concerning the coffee temperature? Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm For the assumed exponential model ln(Temp)= β<sub>0</sub> + β<sub>1</sub>Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the regression equation for making predictions concerning the coffee temperature? Refer to Exhibit 16-4.What is the regression equation for making predictions concerning the coffee temperature?

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Question 75

Exhibit 16.2.Typically,the sales volume declines with an increase of a product price.It has been observed,however,that for some luxury goods the sales volume may increase when the price increases.The following Excel output illustrates this rather unusual relationship. Exhibit 16.2.Typically,the sales volume declines with an increase of a product price.It has been observed,however,that for some luxury goods the sales volume may increase when the price increases.The following Excel output illustrates this rather unusual relationship. Refer to Exhibit 16.2.What can be said about the linear relationship between Price and Sales? Refer to Exhibit 16.2.What can be said about the linear relationship between Price and Sales?

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Question 76

When the data is available on x and y,it is easy to estimate a polynomial regression model.

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Question 77

The fit of the models y = β0 + β1x + β2x2 + ε and y = β0 + β1ln(x)+ ε can be compared using the coefficient of determination R2.

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Question 78

The cubic regression model,y = β0 + β1x + β2x2+ β3x3 + ε,is used when we assume that the relationship between x and y should be captured by a function that has either minimum or maximum,but not both.

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Question 79

Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm For the assumed exponential model ln(Temp)= β<sub>0</sub> + β<sub>1</sub>Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the standard error of the estimate? For the assumed exponential model ln(Temp)= β0 + β1Time + ε,the following Excel regression partial output is available. Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm For the assumed exponential model ln(Temp)= β<sub>0</sub> + β<sub>1</sub>Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the standard error of the estimate? Exhibit 16-4.The following data shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup.The brewing pot temperature is approximately 180º F;see http://mathbits.com/mathbits/tisection/statistics2/exponential.htm For the assumed exponential model ln(Temp)= β<sub>0</sub> + β<sub>1</sub>Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the standard error of the estimate? Refer to Exhibit 16-4.What is the standard error of the estimate?

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