Data babies in openintro package

## Rows: 1,236
## Columns: 8
## $ case      <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, …
## $ bwt       <int> 120, 113, 128, 123, 108, 136, 138, 132, 120, 143, 140, 144,…
## $ gestation <int> 284, 282, 279, NA, 282, 286, 244, 245, 289, 299, 351, 282, …
## $ parity    <int> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ age       <int> 27, 33, 28, 36, 23, 25, 33, 23, 25, 30, 27, 32, 23, 36, 30,…
## $ height    <int> 62, 64, 64, 69, 67, 62, 62, 65, 62, 66, 68, 64, 63, 61, 63,…
## $ weight    <int> 100, 135, 115, 190, 125, 93, 178, 140, 125, 136, 120, 124, …
## $ smoke     <int> 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1,…

Notation

$y_i={\beta }_0+{\beta }_1{x}_i+{ϵ}_i$

${\beta }_0$ is y-intercept
${\beta }_1$ is slope
${ϵ}_i$ is error/residual
$i=1,2,...n$ identifier for each point

model_s <- lm(bwt ~ smoke, data = babies)
tidy(model_s)

## # A tibble: 2 x 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)   123.       0.649    190.   0.      
## 2 smoke          -8.94     1.03      -8.65 1.55e-17

${\hat{y}}_i=b_0+b_1{x}_i$

$\widehat{{\text{bwt}}_i}=b_0+b_1{\text{smoke}}_i$

$\widehat{{\text{bwt}}_i}=123+(-8.94{\text{smoke}}_i)$

confint(model_s)

##                 2.5 %     97.5 %
## (Intercept) 121.77391 124.320430
## smoke       -10.96413  -6.911199

Note that the confidence interval for the "slope" does not contain 0 and all the values in the interval are negative.