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I have to two groups and I would like to test that difference between means of these two groups is at least 0.02. I am using TOSTER R package. Here are details:

library(TOSTER)
TOSTtwo.raw(m1 =0.05, m2 = 0.07,
   sd1 =0.25, sd2 = 0.27, n1 = 10000, n2 = 11000, 
   low_eqbound = - 0.02, high_eqbound = 0.02, alpha = 0.05)

Here is output:

TOST results:
t-value lower bound: -0.000000000000000967  p-value lower bound: 0.500
t-value upper bound: -11.15     p-value upper bound: 0.00000000000000000000000000004
degrees of freedom : 20990.93

Equivalence bounds (raw scores):
low eqbound: -0.02 
high eqbound: 0.02

TOST confidence interval:
lower bound 90% CI: -0.026
upper bound 90% CI:  -0.014

NHST confidence interval:
lower bound 95% CI: -0.027
upper bound 95% CI:  -0.013

Equivalence Test Result:
The equivalence test was non-significant, t(20990.93) = -0.000000000000000967, p = 0.500, given equivalence bounds of -0.020 and 0.020 (on a raw scale) and an alpha of 0.05.

Null Hypothesis Test Result:
The null hypothesis test was significant, t(20990.93) = -5.573, p = 0.0000000253, given an alpha of 0.05.

Based on the equivalence test and the null-hypothesis test combined, we can conclude that the observed effect is statistically different from zero and statistically not equivalent to zero.

My first question is why there are two tests results TOST and NHST?

My second question. Would it be correct for me to conclude that since we fail to reject null hypothesis |m1-m2| >=0.02, then |m1-m2| >=0.02 is true.

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