The null and alternate hypotheses Ho mu1 lessthan or equal t

The null and alternate hypotheses H_o: mu_1 lessthan or equal to mu_2 H_1: mu_1 greater than mu_2 A random sample of 20 items from the first population showed a mean of 108 and a standard deviation of 16. A smaple of 17 items for the second poplulation showed a mean of 103 and a standard deviation of 12. Assume the sample populations do not have equal standard deviations. Find the degrees of freedom for unequal variance test. State decision rule for 0.025 significance level. Compute the value of the test statistic. What is lylour decision regarding the null hypothesis? Use the 0.025 significance level.

Solution

Set Up Hypothesis
Null, u1 <= u2
Alternate, H1: u1 > u2
Test Statistic
X(Mean)=108
Standard Deviation(s.d1)=16 ; Number(n1)=20
Y(Mean)=103
Standard Deviation(s.d2)=12; Number(n2)=17
we use Test Statistic (t) = (X-Y)/Sqrt(s.d1^2/n1)+(s.d2^2/n2)
to =108-103/Sqrt((256/20)+(144/17))
to =1.08
| to | =1.08
Critical Value
The Value of |t | with Min (n1-1, n2-1) i.e 16 d.f is 2.12
We got |to| = 1.08413 & | t | = 2.12
Make Decision
Hence Value of |to | < | t | and Here we Do not Reject Ho
P-Value:Right Tail -Ha : ( P > 1.0841 ) = 0.14719
Hence Value of P0.025 < 0.14719,Here We Do not Reject Ho

[ANSWERS]
1. Min (n1-1, n2-1) i.e 16 d.f OR (n1+n2-2) i.e 35 d.f
2. Reject Ho, if t>2.12 OR Reject Ho, if t>2.03
3. to = 1.08
4. Do not Reject Ho