10.5 200 patients who suffered from major depression were studied for 12 months.
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Question
10.5 200 patients who suffered from major depression were studied for 12 months. The first group of n1 = 98 patients received treatment A and at the end of the study 14 were in remission. The second group of n2 =102 patients received treatment B, a placebo and at the end of the study, 6 were in remission. Let p1 be the proportion of all patients suffering from major depression who are given treatment A and are in remission after 12 months and let p2 be the proportion of all patients suffering from major depression who are given treatment B and are in remission after 12 months.
(a) Give the value of an unbiased estimator of p1 p2.
(b) Calculate a 90% confidence interval for p1 p2.
(c) It is desired to test if treatment A is better than the placebo. State the null and alternative hypothesis in terms of p1 and p2.
(d) With = .05, what conclusion is reached when the hypotheses in part (c) are tested? State your conclusion in the terminology of this example; not just reject some hypothesis or do not reject a hypothesis.
(e) What is the p-value of the test in part (d)? (f) Using your answer in part (e), what conclusion would be reached if = .01?
Explanation / Answer
a.
p1 p2 = 0.142857 - 0.058824 = 0.084033
b.
Confidence Interval for Diffrence of Proportion
CI = (p1 - p2) ± Z a/2 Sqrt(p1(1-p1)/n1 + p2(1-p2)/n2 )
Proportion 1
No. of chances( X1 )=14
No.Of Observed (n1)=98
P1= X1/n1=0.142857
Proportion 2
No. of chances(X2)=6
No.Of Observed (n2)=102
P2= X2/n2=0.058824
C.I = (0.142857-0.058824) ±Z a/2 * Sqrt( (0.142857*0.857143/98) + (0.058824*0.941176/102) )
=(0.142857-0.058824) ± 1.64* Sqrt(0.001792)
=0.084034-0.069429,0.084034+0.069429
=[0.014604,0.153463]
c.
Null, There Is No Significance between them Ho: p1 = p2
Alternate, treatment A is better than the placebo H1: p1 > p2
d.
Test Statistic
Sample 1 : X1 =14, n1 =98, P1= X1/n1=0.143
Sample 2 : X2 =6, n2 =102, P2= X2/n2=0.059
Finding a P^ value For Proportion P^=(X1 + X2 ) / (n1+n2)
P^=0.1
Q^ Value For Proportion= 1-P^=0.9
we use Test Statistic (Z) = (P1-P2)/(P^Q^(1/n1+1/n2))
Zo =(0.143-0.059)/Sqrt((0.1*0.9(1/98+1/102))
Zo =1.98
| Zo | =1.98
Critical Value
The Value of |Z | at LOS 0.05% is 1.645
We got |Zo| =1.98 & | Z | =1.645
Make Decision
Hence Value of | Zo | > | Z | and Here we Reject Ho
P-Value: Right Tail -Ha : ( P > 1.9803 ) = 0.02384
Hence Value of P0.05 > 0.02384,Here we Reject Ho
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