Deck 16: Regression Models for Nonlinear Relationships

Question 1

The regression model ln(y)= β₀ + β₁x + ε is called exponential.

Answer 1

True

Question 2

Many non-linear regression models can be studied under the linear regression framework using transformation of the response variable and/or the explanatory variables.

Answer 2

True

Question 3

If the data is available on the response variable y and the explanatory variable x,and the fit of the quadratic model y = β₀ + β₁x + β₂x² + ε is to be tested,standard linear regression can be applied on:

A)y and x
B)y,x and x²
C)y,xy,and x²
D)y,y² and x²

Answer 3

y,x and x²

Question 4

Although a polynomial regression model of order two or more is nonlinear,when it is fitted to the data we use the _______ regression to make this fit.

A)nonlinear
B)logistic
C)polynomial
D)linear

Question 5

For the logarithmic model y = β₀ + β₁ln(x)+ ε,β₁/100 is the approximate change in E(y)when x increases by one percent.

Question 6

The fit of the models y = β₀ + β₁x + ε and ln(y)= β₀ + β₁x + ε can be compared using the coefficients R² found in the two corresponding Excel's regression outputs.

Question 7

The fit of the regression equations and can be compared using the coefficient of determination R².

Question 8

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

Question 9

For the exponential model ln(y)= β₀ + β₁x + ε,β₁ × 100% is the approximate percentage change in E(y)when x increases by one percent.

Question 10

How many coefficients have to be estimated in the quadratic regression modely = β₀ + β₁x + β₂x² + ε?

A)4
B)3
C)2
D)1

Question 11

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.

Question 12

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

Question 13

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

Question 14

Which of the following is a quadratic regression equation?

A)
B)
C)
D)

Question 15

The curve representing the regression equation has a U-shape if b₂ > 0.

Question 16

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.

Question 17

Which of the following regression models is not polynomial?

A)y = β₀ + β₁x + ε
B)y = β₀ + β₁x + β₂x² + ε
C)y = β₀ + β₁x^-1 + ε
D)y = β₀ + β₁x + β₂x²+ β₃x³ + ε

Question 18

The equation y = β₀ + β₁x + β₂x² + ε is called a cubic regression model.

Question 19

The regression model ln(y)= β₀ + β₁ln(x)+ ε is called logarithmic.

Question 20

For the model ln(y)= β₀ + β₁ln(x)+ ε with 0 < β₁ < 1,if x increases than E(y)increases but at a slower rate.

Question 21

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.For which value of Hires the predicted Productivity is maximized (Do not round to the nearest integer. )?

A)29.58
B)124.60
C)35.086
D)27.34

Question 22

What is the effect of b₂ < 0 in the case of the quadratic equation ?

A)The curve is U-shaped.
B)The curve is inverted U-shaped.
C)The curve is a straight line.
D)The curve is not a parabola.

Question 23

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:

A)concave function.
B)hyperbola.
C)convex function.
D)linear function.

Question 24

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 values of Hires can be non-integers,what is the maximum value of Productivity?

A)29.58
B)124.603
C)35.086
D)127.50

Question 25

For the quadratic regression equation ,the predicted y achieves its optimum (maximum or minimum)when x is:

A)
B)
C)
D)

Question 26

An inverted U-shaped curve is also known to be:

A)concave
B)convex
C)opaque
D)hyperbola

Question 27

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.The quadratic regression equation found is:

A) = 35.086 + 6.0523Hires - 0.1023Hires².
B) = 6.0523 + 35.086Hires - 0.1023Hires².
C) = 6.0523 - 35.086Hires + 0.1023Hires².
D)

Question 28

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.What is the percentage of variations in the productivity explained by the number of hired workers?

A)85.69%
B)0.7342%
C)90.54%
D)73.42%

Question 29

For the quadratic regression equation ,the optimum (maximum or minimum)value of is:

A)
B)
C)
D)

Question 30

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?

A)The relationship is negatively moderate.
B)There is no relationship.
C)The relationship is positively strong.
D)The relationship is negatively strong.

Question 31

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,what is the maximum productivity to achieve?

A)29.58
B)30.00
C)124.603
D)124.585

Question 32

Given the data on y and x,what is needed to run Excel regression for the polynomial model of order 3?

A)Creating the values of one pseudo-explanatory variable by squaring the values of x.
B)Creating the values of two pseudo-explanatory variables by squaring and cubing the values of x,respectively.
C)Creating the values of three pseudo-explanatory variables by raising the values of x to the power of 2,3 and 4,respectively.
D)Nothing is needeD.

Question 33

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?

A)26
B)28
C)30
D)32

Question 34

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 is the number of estimated coefficients of the cubic regression model?

A)1
B)2
C)3
D)4

Question 35

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

A)b₁ + 2b₂x
B)2b₁ + b₂x
C)
D)

Question 36

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.Which of the following models is most likely to be chosen in order to describe the relationship between Price and Sales?

A)Linear
B)Quadratic
C)Cubic
D)Exponential

Question 37

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.Using the quadratic equation,predict the sales if the luxury good is priced at $100.

A)1191.87
B)1157.64
C)1160.79
D)1168.00

Question 38

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.Predict the productivity when 32 workers are hired.

A)124.00
B)122.46
C)121.60
D)113.50

Question 39

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.Using the cubic regression equation,predict the sales if the luxury good is priced at $100.

A)1171.85
B)1133.10
C)1106.61
D)1092.91

Question 40

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

A)the quadratic model has one parameter more to estimate.
B)the quadratic model has two parameters more to estimate.
C)the quadratic model always has a lower R².
D)R² is not defined for the quadratic model.

Question 41

For which of the following models,the formula = exp(b₀ + b₁x + )for finding the predicted value of y is used?

A)y = β₀ + β₁x + ε
B)ln(y)= β₀ + β₁ln(x)+ ε
C)y = β₀ + β₁ln(x)+ ε
D)ln(y)= β₀ + β₁x + ε

Question 42

A model in which the response variable is transformed into its natural logarithm is called a(n)_____.

A)log-log model
B)logarithmic model
C)exponential model
D)linear model

Question 43

When the predicted value of the response variable has to be found,in which of the following two models,is there a need for the standard error correction?

A)Linear and Log-log
B)Log-log and Logarithmic
C)Logarithmic and Linear
D)Log-log and Exponential

Question 44

In the model ln(y)= β₀ + β₁ln(x)+ ε,the coefficient β₁ is the approximate:

A)change in E(y)when x increases by one unit.
B)percentage change in E(y)when x increases by 1%.
C)percentage change in E(y)when x increases by one unit.
D)change in E(y)when x increases by 1%.

Question 45

A model with one explanatory variable being the only one transformed into its natural logarithm is called a(n)_____.

A)log-log model
B)logarithmic model
C)exponential model
D)linear model

Question 46

For the exponential model ln(y)= β₀ + β₁x + ε,if x increases by 1 unit,then E(y)changes by approximately

A)β₁ × 100 percent.
B)β₁ × 100 units.
C)β₁ percent.
D)β₁ units.

Question 47

The linear and logarithmic models,y = β₀ + β₁x + ε and y = β₀ + β₁ln(x)+ ε,were used to fit given data on y and x,and the following table summarizes the regression results.Which of the two models provides a better fit?

A)The linear model.
B)The logarithmic model.
C)The models are not comparable.
D)The provided information is not sufficient to make the conclusion.

Question 48

The logarithmic and log-log models,y = β₀ + β₁ln(x)+ ε and ln(y)= β₀ + β₁ln(x)+ ε,were used to fit given data on y and x,and the following table summarizes the regression results.Which of the two models provides a better fit?

A)The logarithmic model.
B)The log-log model.
C)The models are not comparable.
D)The provided information is not sufficient to make the conclusion.

Question 49

Which of the regression models is most likely to provide the best fit for the data represented by the following scatterplot?

A)exponential model.
B)logarithmic model.
C)linear model.
D)log-log model.

Question 50

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 which price do sales predicted by the quadratic equation reach their minimum?

A)106.33
B)1157.16
C)100.41
D)1166.64

Question 51

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)= β₀ + β₁Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the standard error of the estimate?

A)0.03421
B)0.45476
C)0.00177
D)0.67436

Question 52

For the log-log model ln(y)= β₀ + β₁ln(x)+ ε,the predicted value of y is computed by:

A)
B)
C)
D)

Question 53

A model in which both the response variable and the explanatory variable are transformed into their natural logarithms is better known as a(n):

A)exponential model.
B)logarithmic model.
C)linear model.
D)log-log model.

Question 54

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)= β₀ + β₁Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the sample correlation coefficient between ln(Temp)and Time?

A)-0.9701
B)0.9701
C)-0.9849
D)0.9849

Question 55

The quadratic and logarithmic models,y = β₀ + β₁x + β₂x² + ε and y = β₀ + β₁ln(x)+ ε,were used to fit given data on y and x,and the following table summarizes the regression results.Which of the two models provides a better fit?

A)The quadratic model.
B)The logarithmic model.
C)The models are not comparable.
D)The provided information is not sufficient to make the conclusion.

Question 56

The log-log and exponential models,ln(y)= β₀ + β₁ln(x)+ ε and ln(y)= β₀ + β₁x + ε,were used to fit given data on y and x,and the following table summarizes the regression results.Which of the two models provides a better fit?

A)The log-log model.
B)The exponential model.
C)The models are not comparable.
D)The provided information is not sufficient to make the conclusion.

Question 57

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 which two prices are the sales predicted by the quadratic equation are 1700 units?

A)60.51 and 150.15
B)61.51 and 151.15
C)62.51 and 152.15
D)63.51 and 153.15

Question 58

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)= β₀ + β₁Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the percentage of variations in ln(Temp)explained by Time?

A)45.48%
B)97.01%
C)1.40%
D)46.88%

Question 59

For the logarithmic model y = β₀ + β₁ln(x)+ ε,the predicted value of y is computed by:

A)
B)
C)
D)

Question 60

What does a positive value for price elasticity indicate if y represents the quantity demanded of a particular good and x is its unit price in a log-log regression model?

A)As price increases,the expected sales decreases.
B)As price decreases,the expected sales increases.
C)As price increases,the expected sales increases.
D)As price decreases,the expected sales remain the same.

Question 61

Exhibit 16.6.Thirty employed single individuals were randomly selected to examine the relationship between their age (Age)and their credit card debt (Debt)expressed as a percentage of their annual income.Three polynomial models were applied and the following table summarizes Excel's regression results. Refer to Exhibit 16.6.Using the quadratic regression equation,find the predicted maximum percentage debt.

Question 62

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 = β₀ + β₁Price + β₂Price² + β₃Price³ + ε and ln(Demand)= β₀ + β₁ln(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?

A)98.52%
B)98.50%
C)91.39%
D)97.93%

Question 63

Exhibit 16.6.Thirty employed single individuals were randomly selected to examine the relationship between their age (Age)and their credit card debt (Debt)expressed as a percentage of their annual income.Three polynomial models were applied and the following table summarizes Excel's regression results. Refer to Exhibit 16.6.Using the quadratic regression equation,find the age of an employed single person with the highest predicted percentage debt.

Question 64

Exhibit 16.6.Thirty employed single individuals were randomly selected to examine the relationship between their age (Age)and their credit card debt (Debt)expressed as a percentage of their annual income.Three polynomial models were applied and the following table summarizes Excel's regression results. Refer to Exhibit 16.6.What is the sample correlation coefficient between Age and Debt?

Question 65

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)= β₀ + β₁Time + 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?

A)
B)
C)
D)

Question 66

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)= β₀ + β₁Time + 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 ^oF?

A)About 5 minutes
B)About 6 minutes
C)About 7 minutes
D)About 8 minutes

Question 67

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 = β₀ + β₁Price + β₂Price² + β₃Price³ + ε and ln(Demand)= β₀ + β₁ln(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 percentage of variations in Demand explained by the log-log regression equation?

A)98.52%
B)98.50%
C)91.39%
D)97.93%

Question 68

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)= β₀ + β₁Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.During one minute,the predicted temperature decreases by approximately

A)0.0118 ⁰F.
B)1.18 ⁰F.
C)1.18 %.
D)11.8 %.

Question 69

Exhibit 16.6.Thirty employed single individuals were randomly selected to examine the relationship between their age (Age)and their credit card debt (Debt)expressed as a percentage of their annual income.Three polynomial models were applied and the following table summarizes Excel's regression results. Refer to Exhibit 16.6.What is the estimate of the variance of the random error ε provided by the regression equation with the best fit?

Question 70

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 = β₀ + β₁Price + β₂Price² + β₃Price³ + ε and ln(Demand)= β₀ + β₁ln(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?

A)14378.72
B)9201.45
C)10764.66
D)12499.98

Question 71

Exhibit 16.6.Thirty employed single individuals were randomly selected to examine the relationship between their age (Age)and their credit card debt (Debt)expressed as a percentage of their annual income.Three polynomial models were applied and the following table summarizes Excel's regression results. Refer to Exhibit 16.6.What is the predicted percentage debt of a 45 year old employed single person determined by the model with the best fit?

Question 72

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 = β₀ + β₁Price + β₂Price² + β₃Price³ + ε and ln(Demand)= β₀ + β₁ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.What does the slope of the obtained regression equation signify?

A)For every 1% increase in the price,the predicted demand declines by approximately 3.2577%.
B)For every 1% increase in the demand,the expected price increases by approximately 3.2577%.
C)For every 1% increase in the demand,the expected price decreases by approximately 3.2577%.
D)For every 1% increase in the price,the predicted demand increases by approximately 3.2577%.

Question 73

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 = β₀ + β₁Price + β₂Price² + β₃Price³ + ε and ln(Demand)= β₀ + β₁ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.What is the price elasticity of the demand found by the log-log model?

A)26.3660
B)-3.2577
C)0.9852
D)0.2071

Question 74

Exhibit 16.6.Thirty employed single individuals were randomly selected to examine the relationship between their age (Age)and their credit card debt (Debt)expressed as a percentage of their annual income.Three polynomial models were applied and the following table summarizes Excel's regression results. Refer to Exhibit 16.6.What is the regression equation that provides the best fit?

Question 75

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 = β₀ + β₁Price + β₂Price² + β₃Price³ + ε and ln(Demand)= β₀ + β₁ln(Price)+ ε,the following regression results are available: Refer to Exhibit 16.5.Using the log-log model,what is the predicted demand when the price is $200?

A)10874.92
B)9201.45
C)7849.25
D)12499.98

Question 76

Exhibit 16.6.Thirty employed single individuals were randomly selected to examine the relationship between their age (Age)and their credit card debt (Debt)expressed as a percentage of their annual income.Three polynomial models were applied and the following table summarizes Excel's regression results. Refer to Exhibit 16.6.What is the value of the test statistic for testing H₀: β₂ = β₃ = 0 against H_A: β₂ ≠ 0 or β₃ ≠ 0 in the model Debt = β₀ + β₁Age + β₂Age²+ β₃Age³ + ε?

Question 77

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 = β₀ + β₁Price + β₂Price² + β₃Price³ + ε and ln(Demand)= β₀ + β₁ln(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?

A)4447.88
B)3914.38
C)4029.38
D)5137.60

Question 78

Exhibit 16.6.Thirty employed single individuals were randomly selected to examine the relationship between their age (Age)and their credit card debt (Debt)expressed as a percentage of their annual income.Three polynomial models were applied and the following table summarizes Excel's regression results. Refer to Exhibit 16.6.What is the percentage of variations in Debt explained by Age in the regression equation with the best fit?

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 For the assumed exponential model ln(Temp)= β₀ + β₁Time + ε,the following Excel regression partial output is available. Refer to Exhibit 16-4.What is the predicted coffee temperature in half an hour after the brewing?

A)164.72
B)-4.7904
C)164.74
D)120.42

Question 80

Exhibit 16.6.Thirty employed single individuals were randomly selected to examine the relationship between their age (Age)and their credit card debt (Debt)expressed as a percentage of their annual income.Three polynomial models were applied and the following table summarizes Excel's regression results. Refer to Exhibit 16.6.If you impose the restrictions β₂ = β₃ = 0 on the model Debt = β₀ + β₁Age + β₂Age²+ β₃Age³ + ε,what will be the sum of the squared errors (SSE_R)computed for the restricted model?