14.3 Use ANOVA

The Null hypothesis is: $$H_0: \text{it is true that } \mu_P=\mu_A=\mu_J$$ While the alternative is:

$$H_0: \text{it is NOT true that } \mu_P=\mu_A=\mu_J$$

The Sample Variance of Y: $$Var(Y)=\frac{1}{N}\sum_{k=1}^G\sum_{i=1}^{N_k}(Y_{ik}-\bar{Y})^2$$ ### Example 2

N<- 5 # number of people
G<- 2 # groups

data2 <- tibble(name=c("Ann","Ben","Cat","Dan","Egg"),
       person_p=seq_along(1:5),
       group=c("cool","cool","cool","uncool","uncool"),
       group_k=c(1,1,1,2,2),
       index_i=c(1,2,3,1,2),
       grumpiness_Yp=c(20,55,21,91,22)
       )
data2

## # A tibble: 5 × 6
##   name  person_p group  group_k index_i grumpiness_Yp
##   <chr>    <int> <chr>    <dbl>   <dbl>         <dbl>
## 1 Ann          1 cool         1       1            20
## 2 Ben          2 cool         1       2            55
## 3 Cat          3 cool         1       3            21
## 4 Dan          4 uncool       2       1            91
## 5 Egg          5 uncool       2       2            22

$$Var(Y)=\frac{1}{N}\sum_{p=1}^{N}(Y_{p}-\bar{Y})^2$$