Deck 16: Regression Models for Nonlinear Relationships

Question 1

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

Answer 1

False

Question 2

For the log-log model ln(y) = β₀ + β₁ln(x) + ε, β₁ is the approximate percent change in E(y) when x increases by 1%.

Answer 2

True

Question 3

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

Answer 3

False

Question 4

The cubic regression model allows for ________ change(s) in slope.

Question 5

When not all variables are transformed with logarithms, the models are called ________ models.

Question 6

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 7

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

Question 8

For the quadratic regression model = b₀ + b₁x + b₂x², b₁ can be interpreted as the change in the predicted value of y when x increases by 1 unit.

Question 9

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

Question 10

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

Question 11

The quadratic regression model = b₀ + b₁x + b₂x² reaches a maximum when b₂ < 0.

Question 12

The curve representing the regression equation = b₀ + b₁x + b₂x² has a U-shape if b₂ < 0.

Question 13

The quadratic regression model = b₀ + b₁x + b₂x² allows for one sign change in the slope capturing the influence of x on y.

Question 14

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

Question 15

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

Question 16

The cubic regression model = b₀ + b₁x + b₂x² + b₃x³ allows for one sign change in the slope capturing the influence of x on y.

Question 17

The quadratic regression model allows for one change in ________.

Question 18

A quadratic regression model is a polynomial regression model of order 2.

Question 19

A cubic regression model is a polynomial regression model of order 2.

Question 20

The log-log and the ________ models can have similar shapes.

Question 21

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

A) -2b₂/b_1
B) -b₁/2b_{2
C) b1}/2b_2
D) 2b₁/b₂

Question 22

Which of the following regression models is a first-order polynomial?

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

Question 23

The following scatterplot shows productivity and number hired workers with a fitted quadratic regression model. Assuming that the number of hired workers must be integer, how many workers should be hired to achieve the highest productivity according to the model?

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

Question 24

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

Question 25

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) -b₁/2b_2
D) -b₂/2b₂

Question 26

The following scatterplot shows productivity and number hired workers with a fitted quadratic regression model. Assuming that the number of hired workers must be an integer, what is the maximum productivity predicted by the model?

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

Question 27

The following scatterplot shows productivity and number hired workers with a fitted quadratic regression model. The quadratic regression model is ________.

A) = 35.086 + 6.0523Hires - 0.1023Hires<sup>2

B)</sup>
= 6.0523 + 35.086Hires - 0.1023Hires<sup>2


C)</sup>
= 6.0523 − 35.086Hires + 0.1023Hires<sup>2


D)</sup>
= −0.1023 + 6.0523Hires + 35.086Hires²

Question 28

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

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

Question 29

For a quadratic regression model , it is important to evaluate the estimated ________ effect of the explanatory variable x on the predicted value of the response variable .

Question 30

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 31

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

A) -b₁/2b_{2
B) b1}/2b_2
C) b₀ -
D) b₀ -

Question 32

The following scatterplot shows productivity and number hired workers with a fitted quadratic regression model. Assuming that the values of Hires can be nonintegers, what is the maximum value of Productivity predicted by the model?

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

Question 33

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

Question 34

The following scatterplot shows productivity and number hired workers with a fitted quadratic regression model. 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 35

Although allowing for nonlinear trends, polynomials are still linear in the ________.

A) response
B) explanatory variables
C) coefficients
D) none of these options

Question 36

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

Question 37

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

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

Question 38

Which of the following is a quadratic regression equation?

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

Question 39

The following scatterplot shows productivity and number hired workers with a fitted quadratic regression model. 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 40

The following scatterplot shows productivity and number hired workers with a fitted quadratic regression model. What is percentage of the variation in productivity is explained by the quadratic regression model?

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

Question 41

A model in which both the response variable and the explanatory variable have been log transformed is called a(n) ________.

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

Question 42

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 scatterplot 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 43

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) β₁ units

Question 44

What R function is used to fit a quadratic regression model?

A) lm
B) predict
C) summary
D) cor

Question 45

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 scatterplot 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 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 scatterplot 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

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 48

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 49

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

A) = b₀ + b₁x
B) = exp(b₀ + b₁ln(x) + /2)
C) = b₀ + b₁ln(x)
D) = exp(b₀ + b₁x + /2)

Question 50

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

A) = exp(b₀ + b₁ln(x) + /2)
B) = b₀ + b₁ln(x)
C) = b₀ + b₁x
D) = exp(b₀ + b₁x + /2)

Question 51

A model with one explanatory variable that has been log transformed is called a(n) ________.

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

Question 52

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 scatterplot 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 53

A model in which the response variable has been log transformed is called a(n) ________.

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

Question 54

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 55

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 scatterplot 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 56

For which of the following models is = exp(b₀ + b₁x + /2) used to find the predicted value of y ?

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 57

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 scatterplot 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 58

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 scatterplot 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 59

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 60

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 scatterplot 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 61

The following data show 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. The output for an exponential model, ln(Temp) = β₀ + β₁Time + ε, t is below. How many minutes must elapse after the brewing in order to cool the coffee to 158°F?

A) About five minutes
B) About six minutes
C) About seven minutes
D) About eight minutes

Question 62

The linear and logarithmic models, y = β₀ + β₁x + ε and y = β₀ + β₁ ln(x) + ε, were 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 63

The following data, with the corresponding 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 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 percent of the variation in heights is explained by the model? ________.

A) 6.09
B) 6.10
C) 98.63
D) Can't determine from the given information

Question 65

The quadratic and logarithmic models, y = β₀ + β₁x + β₂x² + ε and y = β₀ + β₁ ln(x) + ε, were 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 66

The following data show 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. The output for an exponential model, ln(Temp) = β₀ + β₁Time + ε, is below. Which of the following is the regression equation for making predictions concerning the coffee temperature?

A) = exp (5.1444 - 0.0118Time
B) = exp (5.1450 - 0.0118Time
C) = 5.1444 - 0.0118Time
D) = 5.1450 - 0.0118Time

Question 67

The following data show 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. The output for an exponential model, ln(Temp) = β₀ + β₁Time + ε, is below. 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 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

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 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 show 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. The output for an exponential model, ln(Temp) = β₀ + β₁Time + ε, is below. During one minute, the predicted temperature decreases by approximately ________.

A) 0.0118° F
B) 1.18° F
C) 1.18%
D) 11.8%

Question 72

The log-log and exponential models, ln(x) = β₀ + β₁ln(x) + ε and ln(y) = β₀ + β₁x + ε, were 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 73

The logarithmic and log-log models, y = β₀ + β₁ln(x) + ε and ln(y) = β₀ + β₁ ln(x) + ε, were 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 comparable.
D) The provided information is not sufficient to make the conclusion.

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. 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 75

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 76

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 77

The following data show 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. The output for an exponential model, ln(Temp) = β₀ + β₁Time + ε, is below. Which of the following is the sample correlation coefficient between ln(Temp) and Time?

A) −0.9701
B) 0.9701
C) −0.9849
D) 0.9849

Question 78

The following data show 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. The output for an exponential model ln(Temp) = β₀ + β₁Time + ε, t is below. Which of the following is the percentage of variation in ln(Temp) explained by the model?

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

Question 79

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 80

The following data show 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. The output for an exponential model, ln(Temp) = β₀ + β₁Time + ε, t is below. Which of the following is the standard error of the estimate?

A) 0.03421
B) 0.45476
C) 0.00117
D) 0.67436