Deck 16: Regression Models for Nonlinear Relationships

Question 1

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

Answer 1

True

Question 2

The curve representing the regression equation = b₀ + b₁x + b₂x²has a U-shape if b₂ > 0.

Answer 2

True

Question 3

For the logarithmic model y = β₀ + β₁ln(x)+ ε,β₁/100 is the approximate change in E(y)when x increases by 1%.

Answer 3

True

Question 4

It is not very informative to start with developing a scatterplot of the response variable against the explanatory variable.

Question 5

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 6

The quadratic regression model is appropriate when the slope,capturing the influence of x on y,changes in magnitude as well as sign.

Question 7

The regression model ln(y)= β₀ + β₁ ln(x)+ ε is called logarithmic.

Question 8

The log-log regression model is ________ in the variables.

Question 9

The fit of the regression equations = b₀ + b₁x + b₂x² and = b₀ + b₁x + b₂x² + b₃x³ can be compared using the coefficient of determination R².

Question 10

The cubic regression model allows for two changes in ______.

Question 11

The fit of the models y = β₀ + β₁x + ε and ln(y)= β₀ + β₁ln(x)+ ε can be compared using the coefficients R² found in the two corresponding Excel's regression outputs.

Question 12

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

Question 13

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 14

It is important to superimpose linear and quadratic trends on the scatterplot using ______.

Question 15

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

Question 16

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

Question 17

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 18

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

Question 19

For the exponential model ln(y)= β₀ + β₁x + ε,β₁ × 100% is the approximate percentage change in E(y)when x increases by 1%.

Question 20

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 21

For the quadratic equation = b₀ + b₁x + b₂x²,which of the following expressions must be zero in order to minimize or maximize the predicted y?

A) b₁ + 2b₂x
B) 2b₁ + b₂x
C) -b₁/2b₂
D) -b₂/2b₂

Question 22

For the quadratic regression equation = b₀ + b₁x + b₂x²,the predicted y achieves its optimum (maximum or minimum)when x is _________.

A) -2b₂/b₁
B) -b₁/2b₂
C) b₁/2b₂
D) 2b₁/b₂

Question 23

When not all variables are transformed into logarithms the models are called _______ models.

Question 24

The following Excel scatterplot with the fitted quadratic regression equation illustrates the observed relationship between productivity and the number of hired workers. Assuming that the values of Hires can be nonintegers,what is the maximum value of Productivity?

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

Question 25

The following Excel scatterplot with the fitted quadratic regression equation illustrates the observed relationship between productivity and the number of hired workers. 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 26

For the quadratic regression equation = b₀ + b₁x + b₂x²,the optimum (maximum or minimum)value of is _________.

A) -b₁/2b₂
B) b₁/2b₂
C)
D)

Question 27

The log-log and the __________ models can allow similar shapes.

Question 28

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 29

To compute the coefficient of determination R² we have to use Excel's ________ function first to derive the correlation between y and .

Question 30

Which of the following is a quadratic regression equation?

A) = b₀ + b₁x^-1 + b₂x²
B) = b₀ + b₁x + b₂y²
C) = b₀ + b₁x^-1 + b₂x^-2
D)

Question 31

The following Excel scatterplot with the fitted quadratic regression equation illustrates the observed relationship between productivity and the number of hired workers. For which value of Hires is the predicted Productivity maximized? Note: Do not round to the nearest integer.

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

Question 32

How many coefficients need to be estimated in the quadratic regression model?

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

Question 33

What is the effect of b₂ < 0 in the case of the quadratic equation = b₀ + b₁x + b₂x²?

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 34

Which of the following regression models is not polynomial?

A) y = β₀ + β₁x + ε
B) y = β₀ + β₁x + β₂ x² + ε
C) y = β₀ + β₁ x^-1 + ε
D) y = β₀ + β₁x + β₂x² + β₃x³ +ε

Question 35

The logarithmic model is especially attractive when only the ____________ variable is better captured in percentages.

Question 36

An inverted U-shaped curve is also known as _______.

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

Question 37

It is important to evaluate the estimated _________ effect of the explanatory variable x on the predicted value of the response variable .

Question 38

The following Excel scatterplot with the fitted quadratic regression equation illustrates the observed relationship between productivity and the number of hired workers. Which of the following is the predicted productivity when 32 workers are hired?

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

Question 39

If the data are available on the response variable y and the explanatory variable x,and the fit of the quadratic regression 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

Question 40

The following Excel scatterplot with the fitted quadratic regression equation illustrates the observed relationship between productivity and the number of hired workers. 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 41

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

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

Question 42

Typically,the sales volume declines with an increase of a product's 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. 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 43

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 44

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. 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 45

The following Excel scatterplot with the fitted quadratic regression equation illustrates the observed relationship between productivity and the number of hired workers. Assuming that the number of hired workers must be an integer,what is the maximum productivity to achieve?

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

Question 46

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. For which of the following two prices are the sales predicted by the quadratic regression equation equal 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 47

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. For which of the following prices do sales predicted by the quadratic regression equation reach their minimum?

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

Question 48

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 49

For which of the following models,the formula for finding the predicted value of y is used?

A)
B)
C)
D)

Question 50

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 51

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. 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 52

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 53

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. 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 54

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

A)
B)
C)
D)

Question 55

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

A)
B)
C)
D)

Question 56

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. 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 57

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

A) β₁×100%
B) β₁×100 units
C) β₁%
D) β₁units

Question 58

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 59

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

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

Question 60

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 61

The following data,with the corresponding Excel scatterplot,show the average growth rate of Weeping Higan cherry trees planted in Washington,DC.At the time of planting,the trees were one year old and were all six feet in height. If the age of a tree increases by 1%,then its predicted height increases by approximately _________.

A) 6.1082%
B) 0.06108%
C) 6.1082 feet
D) 0.061082 feet

Question 62

The following data show 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. Assuming that the sample correlation coefficient between Demand and = exp(26.3660 - 3.2577 ln(Price)+ (0.2071)²/2)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 63

The following data show 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. Which of the following does the slope of the obtained log-log regression equation = 26.3660 - 3.2577 ln(Price)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 64

The following data,with the corresponding Excel scatterplot,show the average growth rate of Weeping Higan cherry trees planted in Washington,DC.At the time of planting,the trees were one year old and were all six feet in height. What is the regression model used to describe the relationship between Height and Age?

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

Question 65

The following data show 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. Using the log-log model,which of the following is the predicted demand when the price is $200?

A) 10,874.92
B) 9,201.45
C) 7,849.25
D) 12,499.98

Question 66

The logarithmic and log-log models,y = β₀ + β₁ln(x)+ ε and ln(y)= β₀ + β₁ ln(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 67

The following data show 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. Which of the following 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 68

Which of the following 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 69

The linear and logarithmic models,y = β₀ + β₁x + ε and y = β₀ + β₁ ln(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 70

The following data,with the corresponding Excel scatterplot,show the average growth rate of Weeping Higan cherry trees planted in Washington,DC.At the time of planting,the trees were one year old and were all six feet in height. If a cherry tree is planted as a one-year-old and six-foot-tall tree,which of the following is the estimated time needed by the tree to reach 16.5 feet in height?

A) About 4 years
B) About 4.5 years
C) About 5 years
D) About 5.5 years

Question 71

The following data,with the corresponding Excel scatterplot,show the average growth rate of Weeping Higan cherry trees planted in Washington,DC.At the time of planting,the trees were one year old and were all six feet in height. Which of the following is the predicted height of an eight-year-old cherry tree that was planted as a one-year-old and six-foot-tall tree?

A) 54.96
B) 42.66
C) 17.04
D) 18.80

Question 72

Which of the following 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 73

The following data,with the corresponding Excel scatterplot,show the average growth rate of Weeping Higan cherry trees planted in Washington,DC.At the time of planting,the trees were one year old and were all six feet in height. The 98.63% of the variations in Height is explained by _______.

A) Height
B) Age
C) ln(Age)
D) ln(Height)

Question 74

The following data,with the corresponding Excel scatterplot,show the average growth rate of Weeping Higan cherry trees planted in Washington,DC.At the time of planting,the trees were one year old and were all six feet in height. Which of the following is the correlation coefficient between Height and ln(Age)?

A) −0.9863
B) 0.9863
C) −0.9931
D) 0.9931

Question 75

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 76

In the model ln(y)= β₀ + β₁ ln(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 77

The following data show 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. Which of the following 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 78

The log-log and exponential models,ln(x)= β₀ + β₁ln(x)+ ε and (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 79

The quadratic and logarithmic models,y = β₀ + β₁x + β₂x² + ε and y = β₀ + β₁ ln(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 80

The following data show 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. Using the cubic model,which of the following is the predicted demand when the price is $200?

A) 14,378.72
B) 9,201.45
C) 10,764.66
D) 12,499.98