118 Good Afternoon I am looking for help answering the quest

118- Good Afternoon, I am looking for help answering the question below. I not only need the answers, but I want to understand how to find the answers. Thank you!

The following results are for two samples, one from Population 1 and the other from Population 2:
from Population 1 : 3 5 7 5
from Population 2 : 8 9 6 5 12
(a) Compute SS₁ and SS₂.
(b) Using the results from Problem 5a, compute the pooled variance estimate.
(c) Using the result from Problem 5b, obtain sXbar1Xbar2.
(d) Test H₀:12=0 against H1:12 <0 (=.05).
(e) Draw final conclusions.

Solution

Set Up Hypothesis
Null, Ho: u1 > u2
Alternative, H1: u1 < u2
Test Statistic
X (Mean)=5; Standard Deviation (s.d1)=1.633
Number(n1)=4
Y(Mean)=8; Standard Deviation(s.d2)=2.7386
Number(n2)=5
Value Pooled variance S^2= (n1-1*s1^2 + n2-1*s2^2 )/(n1+n2-2)
S^2 = (3*2.6667 + 4*7.4999) / (9- 2 )
S^2 = 5.4285
we use Test Statistic (t) = (X-Y)/Sqrt(S^2(1/n1+1/n2))
to=5-8/Sqrt((5.4285( 1 /4+ 1/5 ))
to=-3/1.563
to=-1.9194
| to | =1.9194
Critical Value
The Value of |t | with (n1+n2-2) i.e 7 d.f is 1.895
We got |to| = 1.9194 & | t | = 1.895
Make Decision
Hence Value of | to | > | t | and Here we Reject Ho
P-Value: Left Tail - Ha : ( P < -1.9194 ) = 0.0482
Hence Value of P0.05 > 0.0482,Here we Reject Ho

[ANSWERS]
a.
Standard Deviation (s.d1)=1.633, Standard Deviation(s.d2)=2.7386
b.S^2 = 5.4285
c. Xbar1Xbar2. = -3
d.Reject Ho