key term - Linear Regression

Definition

Linear regression is a statistical technique used to model the linear relationship between a dependent variable and one or more independent variables. It aims to find the best-fitting straight line that describes the association between the variables.

5 Must Know Facts For Your Next Test

1. Linear regression models the relationship between variables as a straight line, making it useful for making predictions and understanding the strength of the association.
2. The slope of the regression line represents the average change in the dependent variable for a one-unit change in the independent variable, holding all other variables constant.
3. The coefficient of determination (R-squared) measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
4. Assumptions of linear regression include linearity, independence, homoscedasticity, normality, and absence of multicollinearity.
5. Linear regression can be extended to multiple independent variables, known as multiple linear regression, to model more complex relationships.

Review Questions

• Explain how linear regression can be used to analyze the relationship between distance from school and student academic performance.
  • Linear regression can be used to model the relationship between a student's distance from school (the independent variable) and their academic performance (the dependent variable). The regression line would show the average change in academic performance for a one-unit change in distance from school, holding all other factors constant. The strength of the relationship would be indicated by the coefficient of determination (R-squared), which measures how much of the variation in academic performance can be explained by distance from school. This analysis could help educators understand the impact of commute time on student learning and identify students who may need additional support due to longer travel distances to school.
• Describe how linear regression can be used to investigate the relationship between textbook cost and student enrollment in a college course.
  • Linear regression can be used to analyze the relationship between the cost of textbooks (the independent variable) and student enrollment in a college course (the dependent variable). The regression line would show the average change in enrollment for a one-unit change in textbook cost, holding all other factors constant. The coefficient of determination (R-squared) would indicate the strength of the relationship, revealing how much of the variation in enrollment can be explained by textbook cost. This analysis could help college administrators understand the impact of textbook affordability on student enrollment and make informed decisions about pricing and resource allocation to ensure accessibility and maximize enrollment.
• Evaluate how linear regression can be used to model the relationship between a vehicle's fuel efficiency (miles per gallon) and its engine displacement (cubic inches).
  • $$ Linear regression can be used to model the relationship between a vehicle's fuel efficiency (the dependent variable) and its engine displacement (the independent variable). The regression line would show the average change in fuel efficiency for a one-unit change in engine displacement, holding all other factors constant. The coefficient of determination (R-squared) would indicate the strength of the relationship, revealing how much of the variation in fuel efficiency can be explained by engine displacement. This analysis could help consumers and manufacturers understand the tradeoffs between engine size and fuel efficiency, informing purchasing decisions and design considerations. The insights gained from this linear regression model could also be used to predict the fuel efficiency of new vehicle models based on their engine specifications. $$

Related terms

Dependent Variable:

The variable that is being predicted or explained in a regression model, also known as the response variable.

Independent Variable:

The variable(s) that are used to predict or explain the dependent variable, also known as the predictor variable(s).

Least Squares Method:

The statistical method used in linear regression to find the line of best fit by minimizing the sum of the squared differences between the observed and predicted values.