Interpreting Results

There are 3 unknowns in a simple linear regression that we are estimating:

• ${\hat{\beta }}_0$
• ${\hat{\beta }}_1$
• ${\hat{\sigma }}_{ϵ}$

Let's talk about what they mean in words

Interpreting Results

$\widehat{\text{Weight}}=\widehat{{\beta }_0}+\widehat{{\beta }_1}\text{Wing Length}+ϵ$

lm(Weight ~ WingLength, data = Sparrows)

## 
## Call:
## lm(formula = Weight ~ WingLength, data = Sparrows)
## 
## Coefficients:
## (Intercept)   WingLength  
##      1.3655       0.4674

Interpreting Results

$\widehat{\text{Weight}}=\widehat{{\beta }_0}+\widehat{{\beta }_1}\text{Wing Length}+ϵ$

lm(Weight ~ WingLength, data = Sparrows)

## 
## Call:
## lm(formula = Weight ~ WingLength, data = Sparrows)
## 
## Coefficients:
## (Intercept)   WingLength  
##      1.3655       0.4674

Interpreting Results

$\widehat{\text{Weight}}=\widehat{{\beta }_0}+\widehat{{\beta }_1}\text{Wing Length}+ϵ$

lm(Weight ~ WingLength, data = Sparrows)

## 
## Call:
## lm(formula = Weight ~ WingLength, data = Sparrows)
## 
## Coefficients:
## (Intercept)   WingLength  
##      1.3655       0.4674

Interpreting Results

$\widehat{\text{Weight}}=\widehat{{\beta }_0}+\widehat{{\beta }_1}\text{Wing Length}+ϵ$

lm(Weight ~ WingLength, data = Sparrows)

## 
## Call:
## lm(formula = Weight ~ WingLength, data = Sparrows)
## 
## Coefficients:
## (Intercept)   WingLength  
##      1.3655       0.4674

Interpreting Results

$\widehat{\text{Weight}}=\widehat{{\beta }_0}+\widehat{{\beta }_1}\text{Wing Length}+ϵ$

lm(Weight ~ WingLength, data = Sparrows)

## 
## Call:
## lm(formula = Weight ~ WingLength, data = Sparrows)
## 
## Coefficients:
## (Intercept)   WingLength  
##      1.3655       0.4674

Interpreting Results

Sparrows %>%
  mutate(y_hat = lm(Weight ~ WingLength, data = Sparrows) %>% predict(),
         residuals_2 = (Weight - y_hat)^2) %>%
  summarise(rse = sqrt(sum(residuals_2) / (n() - 2)))

##        rse
## 1 1.399595

Interpreting Results

##        rse
## 1 1.399595

y_hat <- lm(Weight ~ WingLength, data = Sparrows) %>% 
  predict()
Sparrows %>%
  mutate(residual = Weight - y_hat) %>%
  select(Weight, residual) %>%
  slice(1:5)

##   Weight    residual
## 1   14.9 -0.02020496
## 2   15.0 -0.85501292
## 3   14.3  1.24941095
## 4   17.0  2.07979504
## 5   16.0  0.61239106