regression

Question 1

What is simple linear regression?

Answer 1

• predict the value of one variable (dependent variable, Y) based on the value of another variable (independent variable, X).
• • It fits a line of best fit through the data points to model the relationship

🔹 Purpose
- Allows prediction of outcomes (Y) from known values of predictors (X).
- Helps quantify how much change in X is associated with change in Y.

Question 2

How does Regression differ from correlation?

Answer 2

• Correlation: Measures strength and direction of a linear relationship (no prediction).
• • Correlation gives a single number r.𓂃˖˳·˖ ִֶָ ⋆🌷͙⋆ ִֶָ˖·˳˖𓂃 ִֶָ
• Regression: Builds a predictive equation to estimate Y from X.
• Regression gives an equation with slope and intercept.

Question 3

What is the regression equation?

Answer 3

Ŷ=a+bX
-Ŷ= Predicted value of the dependent variable
- a= Intercept (value of Y when X = 0)
- b= Slope (change in Y for each one‑unit change in X)
- X= Independent variable

looking at R data

##(intercept) estimate (8.8) --- y-intercept
## age - 0.017 ---- slope
## -------
## multiple R: squared 0.00146

interpreting R
ex. R_2 = 0.0 Age tells us nothing about support (there is no relationship)

!predict support 18 yrs old
Ŷ= 8.84 + (-0.017)(18) = 8.52

Question 4

What do the slope and intercept represent in the regression equation?

Answer 4

• Slope (b):
• Positive → As X increases, Y increases.
• Negative → As X increases, Y decreases.
• Zero → No linear relationship (X does not predict Y).
• Intercept (a):
• Predicted value of Y when X = 0.
• Sometimes meaningful (e.g., baseline score), sometimes not (depends on context).

Question 5

How do you interpret the slope and intercept of a regression equation?

Answer 5

Typical R output includes:
- Intercept (a): The baseline predicted value.
- Slope (b): The rate of change in Y per unit change in X.
- p‑value for slope: Tests whether slope is significantly different from 0.
- R‑squared (R²): Proportion of variance in Y explained by X.
- Residual standard error: Average distance between observed and predicted values.

Example:
- Intercept = 6.42
- Slope = 0.44 (positive, significant, p < .001)
- R² = 0.117 → About 12% of variation in Y explained by X.

Question 6

What does R-squared represent?

Answer 6

R² measures how well the regression model explains variability in Y.

• Range: 0 to 1.
• 0 → Model explains none of the variability.
• 1 → Model explains all variability.
• Interpretation: Higher R² = better fit, but “good” depends on context and field norms.

Question 7

How do you calculate the predicted value in regression?

Answer 7

Steps:
- Write regression equation: Ŷ=a+bX.
- Plug in the given X value.
- Solve for Ŷ.
Example:
Equation:Ŷ=28.8+0.42X
- If X=86:
Ŷ=28.8+0.42(86)=65
Predicted attraction score = 65.