Question

Consider the samples below. Test the following hypothesis at 5% level of significance and find the associated p-value with the test statistic.

μ₁ - μ₂ = 0

μ₁ - μ₂ ≠ 0

Sample 1	Sample 2
725	253
892	665
680	95
127	725
159	541
480	522
579	384
774	670
780	62
517	328
662	277
288	492
986	312
706	80
608	12
867	14
36	747
68	976
735	487
577	30
719	41

Answer 1

assuming independent t test:

Treatment 1

N₁: 21
df₁ = N - 1 = 21 - 1 = 20
M₁: 569.76
SS₁: 1557511.81
s²₁ = SS₁/(N - 1) = 1557511.81/(21-1) = 77875.59


Treatment 2

N₂: 21
df₂ = N - 1 = 21 - 1 = 20
M₂: 367.29
SS₂: 1654170.29
s²₂ = SS₂/(N - 1) = 1654170.29/(21-1) = 82708.51


T-value Calculation

s²_p = ((df₁/(df₁ + df₂)) * s²₁) + ((df₂/(df₂ + df₂)) * s²₂) = ((20/40) * 77875.59) + ((20/40) * 82708.51) = 80292.05

s²_M1 = s²_p/N₁ = 80292.05/21 = 3823.43
s²_M2 = s²_p/N₂ = 80292.05/21 = 3823.43

t = (M₁ - M₂)/√(s²_M1 + s²_M2) = 202.48/√7646.86 = 2.32

Significance Level:

The t-value is 2.31543. The p-value is .025801