10.5 200 patients who suffered from major depression were studied for 12 months.
ID: 3155658 • Letter: 1
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)
Getting p1^ and p2^,
p1^ = x1/n1 = 0.142857143
p2^ = x2/n2 = 0.058823529
Hence,
p1^-p2^ = 0.084033613 [ANSWER]
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b)
Getting p1^ and p2^,
p1^ = x1/n1 = 0.142857143
p2 = x2/n2 = 0.058823529
Also, the standard error of the difference is
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] = 0.042335057
For the 90% confidence level, then
alpha/2 = (1 - confidence level)/2 = 0.05
z(alpha/2) = 1.644853627
Margin of error = z(alpha/2)*sd = 0.069634972
lower bound = p1^ - p2^ - z(alpha/2) * sd = 0.014398641
upper bound = p1^ - p2^ + z(alpha/2) * sd = 0.153668585
Thus, the confidence interval is
( 0.014398641 , 0.153668585 ) [ANSWER]
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c)
Formulating the hypotheses
Ho: p1 - p2 <= 0
Ha: p1 - p2 > 0 [ANSWER]
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d)
Here, we see that pdo = 0 , the hypothesized population proportion difference.
Getting p1^ and p2^,
p1^ = x1/n1 = 0.142857143
p2 = x2/n2 = 0.058823529
Also, the standard error of the difference is
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] = 0.042335057
Thus,
z = [p1 - p2 - pdo]/sd = 1.984965168
As alpha = 0.05, right tailed,
zcrit = 1.645
Also, the p value is, as this is right tailed,
P = 0.023574174
As z > 1.645, P < 0.05, then we REJECT THE NULL HYPOTHESIS.
Hence, there is significant evidence that treatment A is better than the placebo interms of the proportion of those in remission at 0.05 level. [CONCLUSION]
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e)
As said above,
Pvalue = 0.023574174 [ANSWER]
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f)
We do not reject Ho as P > 0.01.
Hence, there is no significant evidence that treatment A is better than the placebo interms of the proportion of those in remission at 0.01 level. [CONCLUSION]
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