Linear Regression Quizzes: Test Your Understanding

Question 1

Which of the following is NOT a fundamental assumption of linear regression?

Accepted Answer

The response variable is normally distributed.

Answer

The errors have constant variance (homoscedasticity).

Answer

The relationship between the variables is linear.

Answer

The errors are independent.

Question 2

In a simple linear regression model, what does the slope coefficient (β1) represent?

Accepted Answer

The average change in the response variable for a one-unit increase in the predictor variable.

Answer

The value of the response variable when the predictor variable is zero (represented by the y-intercept).

Answer

The strength and direction of the relationship between the variables (represented by the correlation coefficient).

Question 3

Which method is most commonly used to estimate the parameters of a linear regression model?

Accepted Answer

Ordinary Least Squares (OLS)

Answer

Method of Moments

Answer

Maximum Likelihood Estimation (MLE)

Answer

Bayesian Estimation

Question 4

A high R-squared value in a linear regression model indicates that:

Accepted Answer

A large proportion of the variance in the response variable is explained by the predictor variable.

Answer

There is a strong causal relationship between the variables (correlation does not imply causation).

Answer

The model is a perfect fit for the data (perfect fit is rarely achieved in real-world data).

Question 5

What is the primary purpose of residual analysis in linear regression?

Accepted Answer

To check the validity of the model assumptions.

Answer

To calculate the predicted values of the response variable (this is a part of the regression process itself).

Answer

To estimate the parameters of the model (this is done by OLS or other estimation methods).

Answer

To determine the significance of the predictor variable (this is determined through hypothesis testing).

Question 6

In a hypothesis test for a regression coefficient, what does a p-value less than the significance level (e.g., 0.05) indicate?

Accepted Answer

There is sufficient evidence to reject the null hypothesis that the coefficient is zero.

Answer

The model is a good fit for the data (p-value only assesses the significance of the coefficient, not overall model fit).

Answer

The coefficient is practically significant (statistical significance does not always imply practical importance).

Question 7

Which of the following is NOT a common transformation technique used to address violations of linear regression assumptions?

Accepted Answer

Exponential smoothing (this is a time series forecasting method, not a transformation for linear regression assumptions).

Answer

Box-Cox transformation (a widely used transformation for non-normal data).

Answer

Square root transformation (useful for stabilizing variance).

Answer

Logarithmic transformation (can address non-linearity and heteroscedasticity).

Question 8

What is the primary purpose of using interaction terms in multiple linear regression?

Accepted Answer

To model the combined effect of two or more predictor variables on the response variable.

Answer

To reduce the number of predictor variables in the model (interaction terms actually increase the number of variables).

Answer

To control for confounding variables (while interaction terms can help account for interactions, they are not the primary method for controlling confounding).

Question 9

Which evaluation metric is commonly used to assess the predictive performance of a linear regression model on new data?

Accepted Answer

Root Mean Squared Error (RMSE)

Answer

Adjusted R-squared (similar to R-squared, focuses on goodness of fit).

Answer

R-squared (primarily measures goodness of fit, not predictive performance on new data).

Answer

Akaike Information Criterion (AIC) (used for model selection, not directly for predictive performance).

Question 10

Which of the following is NOT a core assumption of linear regression?

Accepted Answer

The residuals are normally distributed

Answer

The relationship between the dependent and independent variables is linear

Answer

The errors have constant variance (homoscedasticity)

Answer

The independent variables are independent of each other

Question 11

What method is commonly used to fit a linear model to data in linear regression?

Accepted Answer

Least squares

Answer

Bayesian inference

Answer

Regression trees

Answer

Maximum likelihood estimation

Question 12

In a linear regression model, what does the slope represent?

Accepted Answer

The change in the dependent variable for a one-unit change in the independent variable

Answer

The correlation coefficient between the dependent and independent variables

Answer

The predicted value of the dependent variable when the independent variable is zero

Question 13

What is the primary function of the intercept in a linear regression equation?

Accepted Answer

It represents the predicted value of the dependent variable when the independent variable is zero.

Answer

It estimates the variance of the residuals.

Answer

It adjusts the slope of the regression line.

Question 14

Which metric is commonly used to assess the goodness of fit of a linear regression model?

Accepted Answer

R-squared

Answer

Adjusted R-squared

Answer

Mean squared error (MSE)

Answer

Akaike information criterion (AIC)

Question 15

What is the primary purpose of a scatterplot in linear regression analysis?

Accepted Answer

To visually examine the relationship between the dependent and independent variables.

Answer

To calculate the correlation coefficient.

Answer

To estimate the slope of the regression line.

Question 16

Which transformation can be applied to linearize a nonlinear relationship in regression analysis?

Accepted Answer

Logarithmic transformation

Answer

Square root transformation

Answer

Exponential transformation

Answer

Polynomial transformation

Question 17

What is the key difference between simple linear regression and multiple linear regression?

Accepted Answer

Simple linear regression involves one dependent variable and one independent variable, while multiple linear regression uses one dependent variable and two or more independent variables.

Answer

Simple linear regression assumes a linear relationship, while multiple linear regression assumes a nonlinear relationship.

Question 18

Which of the following is a common application of linear regression?

Accepted Answer

Predicting future sales based on historical data

Answer

Classifying documents into categories

Answer

Clustering customers into segments

Answer

Generating random numbers

Question 19

What are some of the limitations of linear regression?

Accepted Answer

It assumes a linear relationship between variables, can be sensitive to outliers, and may not handle missing data effectively.

Answer

It requires a large sample size.

Question 20

Which of the following IS an assumption of linear regression?

Accepted Answer

The relationship between the variables is linear.

Answer

The errors are normally distributed.

Answer

The errors have constant variance.

Question 21

What does the coefficient of determination (R-squared) represent in a linear regression model?

Accepted Answer

The proportion of variance in the dependent variable explained by the independent variable.

Answer

The correlation between the variables.

Answer

The slope of the regression line.

Question 22

Given a regression equation of Y = 2.5X + 10, what is the predicted value of Y when X = 5?

Accepted Answer

22.5

Answer

12.5

Answer

15

Question 23

Which of the following statistical tests is used to determine the overall significance of a linear regression model?

Accepted Answer

F-test

Answer

Z-test

Answer

T-test

Answer

Chi-square test

Question 24

What is the difference between simple linear regression and multiple linear regression?

Accepted Answer

Simple linear regression models the relationship between one independent variable and one dependent variable, while multiple linear regression models the relationship between two or more independent variables and one dependent variable.

Answer

Simple linear regression uses categorical variables, while multiple linear regression uses continuous variables.

Question 25

A company is trying to model the relationship between advertising expenditure and sales revenue. What type of regression analysis would be most appropriate?

Accepted Answer

Linear regression

Answer

Time series analysis

Answer

Logistic regression

Answer

Cluster analysis

Question 26

What is the purpose of residual analysis in linear regression?

Accepted Answer

To check the assumptions of the model and identify potential outliers.

Answer

To calculate the coefficient of determination (R-squared).

Answer

To determine the significance of the regression coefficients.

Question 27

Which of the following is a potential problem with multicollinearity in multiple linear regression?

Accepted Answer

It can inflate the standard errors of the coefficients.

Answer

It can lead to unstable regression coefficients.

Answer

It makes it easier to interpret the individual regression coefficients.

Question 28

A researcher finds a strong positive correlation between the number of hours studied and exam scores. What can be concluded from this finding?

Accepted Answer

There is a strong association between the variables, but it does not prove causation.

Answer

There is no relationship between the variables.

Answer

Studying more hours directly causes higher exam scores.

Question 29

Which of the following is a common method for dealing with outliers in linear regression?

Accepted Answer

Transforming the data or removing outliers.

Answer

Adjusting the regression coefficients to account for outliers.

Answer

Ignoring the outliers.

Answer

Adding more data points to compensate for outliers.