A study examined the effectiveness of Botox injections in the corrugator superci
ID: 3223921 • Letter: A
Question
A study examined the effectiveness of Botox injections in the corrugator supercilii muscles for the treatment of chronic migraines. Of the 29 migraine-prone patients participating in the study, 16 reported complete migraine elimination. (a) Determine a plus four 90% confidence interval for the population proportion of migraine patients with complete migraine remission following such Botox injections. (b) How large a sample n would you need to estimate p with margin of error 0.02 with 90% confidence? Use p = 0.55 as an estimated value for p. (c) How large a sample n would you need to estimate p with margin of error 0.02 with 90% confidence? Use p = 0.50 as the estimated value for p.Explanation / Answer
(a)
n = 29
p = 16/29 = 0.551724138
% = 90
Standard Error, SE = {p(1 - p)/n} = (0.551724137931034(1 - 0.551724137931034))/29 = 0.092349526
z- score = 1.644853627
Width of the confidence interval = z * SE = 1.64485362695147 * 0.0923495259878602 = 0.15190145
Lower Limit of the confidence interval = P - width = 0.551724137931034 - 0.151901452768381 = 0.39982269
Upper Limit of the confidence interval = P + width = 0.551724137931034 + 0.151901452768381 = 0.70362559
The 90% confidence interval is [0.3998, 0.7036]
(b) z- score for 90% confidence is z = 1.6449
N = (z/E)^2 * pq = (1.6449/0.02)^2 * 0.55 * (1 - 0.55) = 1675
(c) N = (z/E)^2 * pq = (1.6449/0.02)^2 * 0.50 * (1 - 0.50) = 1691.
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