4 The following data recorded in days represent the length o

4. The following data, recorded in days, represent the length of time to recovery for patients randomly treated with one of two medications to clear up severe bladder infections. Assume that the data come from normal populations with equal variance. Test the hypothesis that the two population means are the same against the alternative that the first mean is smaller. Use 5% and 1% significance levels.

Solution

Set Up Hypothesis
Null Hypothesis, There Is NoSignificance between them Ho: u1 > u2
Alternative Hypothesis, There Is Significance between themH1: u1 < u2
Test Statistic
X (Mean)=17; Standard Deviation (s.d1)=1.5
Number(n1)=14
Y(Mean)=19; Standard Deviation(s.d2)=1.8
Number(n2)=16
Value Pooled variance S^2= (n1-1*s1^2 + n2-1*s2^2 )/(n1+n2-2)
S^2 = (13*2.25 + 15*3.24) / (30- 2 )
S^2 = 2.7804
we use Test Statistic (t) = (X-Y)/Sqrt(S^2(1/n1+1/n2))
to=17-19/Sqrt((2.7804( 1 /14+ 1/16 ))
to=-2/0.6102
to=-3.2775
| to | =3.2775
Critical Value ( at 0.05)
The Value of |t ?| with (n1+n2-2) i.e 28 d.f is 1.7
We got |to| = 3.2775 & | t ? | = 1.7
Make Decision
Hence Value of | to | > | t ?| and Here we Reject Ho
P-Value: Left Tail - Ha : ( P < -3.2775 ) = 0.0014
Hence Value of P0.05 > 0.0014,Here we Reject Ho

We claim that First is smaller
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at 0.01 LOS
Critical Value
The Value of |t ?| with (n1+n2-2) i.e 28 d.f is 2.47
We got |to| = 3.2775 & | t ? | = 2.47
Make Decision
Hence Value of | to | > | t ?| and Here we Reject Ho
P-Value: Left Tail - Ha : ( P < -3.2775 ) = 0.0014
Hence Value of P0.01 > 0.0014,Here we Reject Ho

We claim that First is smaller