Question 1

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

Accepted Answer

A quadratic regression model is a specific case of a polynomial regression model where the highest degree of the polynomial is 2.

Question 2

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

Accepted Answer

A U-shaped curve in regression analysis is typically characterized by a relationship between the predictor variable (x) and the response variable (y) that first increases, reaches a peak, and then decreases. The coefficient of the quadratic term (x^2) reflects the magnitude and direction of this U-shape. If the coefficient of the quadratic term (b₂) is positive, then the curve will have a U-shape.

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%.

Accepted Answer

Since the coefficient β₁ is multiplied by ln(x), a 1% increase in x would result in approximately a β₁/100 increase in the predicted value of y.

Question 4

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

Accepted Answer

Developing a scatterplot of the response variable against the explanatory variable is a common first step in exploring the relationship between two variables. The scatterplot can provide visual insight into the nature of the relationship, such as whether it is linear or nonlinear, and whether there are any outliers or patterns in the data. Therefore, it can be informative to start with a scatterplot.

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.

Accepted Answer

The statement is incorrect. The cubic regression model can capture both the minimum and maximum points in the relationship between x and y, depending on the values of the coefficients.

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.

Accepted Answer

A quadratic regression model includes a quadratic term, which allows for changes in the slope's magnitude and sign as x increases.

Question 7

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

Accepted Answer

The given model is called a log-log model, not just logarithmic. It involves taking the natural logarithm of both the dependent variable y and the independent variable x.

Question 8

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

Accepted Answer

The answer of The log-log regression model is ________ in...

Question 9

The fit of the regression equations = b₀ + b₁x + b₂x² and = b₀ + b₁x + b₂x² + b₃x³can be compared using the coefficient of determination R².

Accepted Answer

The answer of The fit of the regression equations ...

Question 10

The cubic regression model allows for two changes in ______.

Accepted Answer

The answer of The cubic regression model allows for two...

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.

Accepted Answer

The R² values from these models are not directly comparable because they are based on different transformations of the dependent variable.

Question 12

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

Accepted Answer

The equation given is actually a polynomial regression model with terms up to degree 3 (cubic), not just 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.

Accepted Answer

The coefficient of determination, or R-squared, measures the proportion of variance in the dependent variable that is predictable from the independent variable(s). It is a statistic used to compare the fit of different regression models, but it cannot be used to directly compare models with different independent variables or functional forms. Therefore, it is not appropriate to compare the fit of the two models in the question using the coefficient of determination.

Question 14

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

Accepted Answer

The answer of It is important to superimpose linear and...

Question 15

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

Accepted Answer

The presence of the natural logarithm of y on the left-hand side of the equation and the presence of the exponentiated terms on the right-hand side make this a form of exponential regression model.

Question 16

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

Accepted Answer

Polynomial regression models can be estimated when the data for the independent variable (x) and dependent variable (y) are available.

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.

Accepted Answer

Because the coefficient β₁ is less than 1, the increase in ln(x) will result in a proportional increase in ln(y) but at a slower rate. Therefore, as x increases, E(y) will increase but at a slower rate than the increase in x.

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.

Accepted Answer

This statement is true. Nonlinear relationships between variables can often be modeled using linear regression methods by transforming either the response variable or the explanatory variables. For example, a logarithmic or power transformation may be applied to one or both variables to make the relationship linear.

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%.

Accepted Answer

The correct formula for the percentage change in E(y) when x increases by 1% is β₁ × 0.01 × 100%, not β₁ × 100%. Therefore, the statement in the question is false.

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.

Accepted Answer

The answer of The fit of the models y =...

Quiz 16: Regression Models for Nonlinear Relationships

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

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

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

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

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.

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

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

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

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

The cubic regression model allows for two changes in ______.

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.

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

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.