1 Null/Alt

\[ n_1=28 \]

\[ n_2=22 \]

I will use function mean.

I am given -

n1 <- 28
n2 <- 22

x1_bar <- 1025
x2_bar <- 910

s1 <- 150
s2 <- 125

2 Method 1: test stat vs critical value

I need to compute my test statistic and compare it with my critical value

2.1 Test Statistic

se <- sqrt( (s1^2 / n1)  + (s2^2 / n2)  )
se
## [1] 38.90757
test_stat <- ((x1_bar - x2_bar) - 0)/se
test_stat
## [1] 2.955723

2.2 Critical Value

df_n <- se^4
df_d <- ( s1^2/n1 )^2/( n1-1 ) + ( s2^2/n2 )^2/( n2-1 )
df <- df_n/df_d

alpha <- .05

critical_value <- 
qt(p          = alpha/2, 
   df         = df, 
   lower.tail = FALSE
   )

Reject the null that no difference between the two means.

3 Method 2: alpha and p value

alpha 
## [1] 0.05
p_value <- 2 * pt(q = test_stat, 
                  df = df, 
                  lower.tail = FALSE
                  )
p_value < alpha
## [1] TRUE

4 Method 3: CONFIDENCE INTERVAL

CI should not include the hypothesized value of 0.

MOE <- qt(p = 1-alpha/2, 
          df = df) * se 
  
upper_bound <- (x1_bar - x2_bar) + MOE
lower_bound <- (x1_bar - x2_bar) - MOE

lower_bound
## [1] 36.76286
upper_bound
## [1] 193.2371