Untitled Flashcards Set

• Population Mean (μ): The true average of a variable in the entire population.

• Sample Mean (X̄): The average computed from a sample, used to estimate the population mean.

• Conditional Expectation (E(Y|X)): The expected value of a variable Y given a certain value of X.

• Regression Equation: Y=β0+β1X+uY = β_0 + β_1X + uY=β0​+β1​X+u, where:

  • YYY = Dependent Variable

  • XXX = Independent Variable

  • uuu = Error Term

  • β0β_0β0​ = Intercept

  • β1β_1β1​ = Slope

• Residual ( u^\hat{u}u^ ): The difference between observed and predicted values (Y−Y^Y - \hat{Y}Y−Y^).

• Ordinary Least Squares (OLS): A method to estimate regression parameters by minimizing the sum of squared residuals.

• Standard Error of Regression (SER): Measures the average size of the residuals.

• R-Squared (R2R^2R2): Measures the proportion of variance in YYY explained by XXX.

Concepts in Flashcard Format

Conditional Expectation

• Q: What is the conditional expectation formula in a simple regression model?

• A: E(Y∣X)=β0+β1XE(Y |X) = β_0 + β_1XE(Y∣X)=β0​+β1​X.

Estimating Regression Parameters

• Q: How do we estimate β0β_0β0​ and β1β_1β1​ in a regression?

• A: Using OLS, we minimize the sum of squared residuals to find:

  • β1^=Cov(X,Y)Var(X)\hat{β_1} = \frac{Cov(X, Y)}{Var(X)}β1​^​=Var(X)Cov(X,Y)​

  • β0^=Yˉ−β1^Xˉ\hat{β_0} = \bar{Y} - \hat{β_1} \bar{X}β0​^​=Yˉ−β1​^​Xˉ

Interpreting Regression Coefficients

• Q: How do you interpret β0β_0β0​ (intercept) in a regression equation?

• A: It is the expected value of YYY when X=0X = 0X=0, but may not always be meaningful if X=0X = 0X=0 is not a realistic scenario.

• Q: How do you interpret β1β_1β1​ (slope) in a regression equation?

• A: It represents the expected change in YYY for a one-unit increase in XXX.

Regression Fit

• Q: What is the Sum of Squared Residuals (SSR)?

• A: The sum of squared differences between observed and predicted values.

• Q: What is Total Sum of Squares (TSS)?

• A: The total variation in the dependent variable (YYY).

• Q: What is Explained Sum of Squares (ESS)?

• A: The portion of TSS explained by the regression model.

• Q: How is R2R^2R2 calculated?

• A: R2=ESSTSS=1−SSRTSSR^2 = \frac{ESS}{TSS} = 1 - \frac{SSR}{TSS}R2=TSSESS​=1−TSSSSR​.

Goodness of Fit

• Q: What does an R2R^2R2 value close to 1 indicate?

• A: A high proportion of the variance in YYY is explained by XXX, meaning a good model fit.

• Q: What does an R2R^2R2 value close to 0 indicate?

• A: The model explains very little of the variance in YYY.

Choosing the Best Regression Line

• Q: What criterion is used to determine the best regression line?

• A: The one that minimizes the sum of squared residuals (OLS method).