Untitled Deck

Question 1

What is the SLR model equation?

Answer 1

y = β₀ + β₁x + ε

Defines the relationship between response variable y and explanatory variable x

Question 2

What is the SLR model for multiple observations?

Answer 2

yᵢ = β₀ + β₁xᵢ + εᵢ

For observations i = 1, 2, …, n

Question 3

What is the objective of Least Squares Estimation (LSE)?

Answer 3

SS(β₀, β₁) = ∑ᵢ=1ⁿ (yᵢ - (β₀ + β₁xᵢ))²

Minimizes the sum of squared residuals

Question 4

How is the slope (Beta 1) estimated in linear regression?

Answer 4

β̂₁ = Sₓᵧ / Sₓₓ = ∑ᵢ=1ⁿ (xᵢ - 𝑥̄)(yᵢ - 𝑦̄) / ∑ᵢ=1ⁿ (xᵢ - 𝑥̄)²

Slope estimator in linear regression

Question 5

What is the formula for estimating the intercept (Beta 0)?

Answer 5

β̂₀ = 𝑦̄ - β̂₁𝑥̄

Intercept estimator in linear regression

Question 6

What is the alternate form for estimating the slope (Beta 1)?

Answer 6

β̂₁ = (∑ᵢ=1ⁿ xᵢyᵢ - n𝑥̄𝑦̄) / (∑ᵢ=1ⁿ xᵢ² - n𝑥̄²)

Alternate form for calculating the slope

Question 7

How can the slope (Beta 1) be calculated using correlation?

Answer 7

β̂₁ = r × (sᵧ / sₓ)

Where r = correlation between x and y

Question 8

What is the formula for calculating sample standard deviation for x?

Answer 8

sₓ = √(1/(n-1) ∑ᵢ=1ⁿ (xᵢ - 𝑥̄)²)

Sample standard deviation for x

Question 9

What is the formula for calculating sample standard deviation for y?

Answer 9

sᵧ = √(1/(n-1) ∑ᵢ=1ⁿ (yᵢ - 𝑦̄)²)

Sample standard deviation for y

Question 10

What is the formula for the sample correlation coefficient (r)?

Answer 10

r = Sₓᵧ / √(Sₓₓ Sᵧᵧ) = ∑ᵢ=1ⁿ (xᵢ - 𝑥̄)(yᵢ - 𝑦̄) / √(∑ᵢ=1ⁿ (xᵢ - 𝑥̄)² ∑ᵢ=1ⁿ (yᵢ - 𝑦̄)²)

Sample correlation coefficient

Question 11

What is the no-intercept SLR model (Beta estimator)?

Answer 11

β̂ = ∑ xᵢyᵢ / ∑ xᵢ²

Used when the regression line passes through the origin