% Proportion of Hemorrhagic-type Strokes Among American Indians.
%
% The study described in the American Heart Association news release of September 22, 2008
% included 4,507 members of 13 American Indian tribes in Arizona, Oklahoma,
% and North and South Dakota.
% It found that American Indians have a stroke rate of 679 per 100,000, compared
% to 607 per 100,000 for African Americans and 306 per 100,000 for Caucasians.
% None of the participants, ages 45 to 74, had a history of stroke when they were recruited
% for the study from 1989 to 1992. Almost 60% of the volunteers were women.
%
% During more than 13 years of follow-up, 306 participants suffered a first stroke,
% most of them in their mid-60s when it occurred.
% There were 263 strokes of the ischemic type -- caused by a blockage that cuts
% off the blood supply to the brain -- and 43 hemorrhagic (bleeding) strokes.
%
% It is believed that in the general population one in five of all strokes is hemorrhagic.
%
% (a) Test the hypothesis that the proportion of hemorrhagic strokes in the
% population of American Indians that suffered a stroke is lower than the
% national proportion of 0.2. Use alpha = 0.05.
%
%
% (b) What is the power of the test in (a) against the alternative
% H_1: p = 0.15?
%
% (c) What sample size ensures power of 90%
% in detecting p=0.15 in a test of level alpha=0.05, if H_0 states
% p = 0.2.
%(a)
phat = 43/306 ; %0.1405
z = (43/306 - 0.2)/sqrt(0.2 *(1- 0.2)/306) %-2.6011
pval = normcdf(z) %0.0046
%Exact p-value
pval = binocdf(43, 306, 0.2) %0.0044
% (b)
p0=0.2; p1=0.15; alpha=0.05; n=306;
power = normcdf( sqrt(p0*(1-p0)/(p1*(1-p1))) * ...
(norminv(alpha) + abs(p1-p0)*sqrt(n)/sqrt( p0*(1-p0)) ) )
%0.728
%(c)
beta = 0.1;
n = p0*(1-p0) * (norminv(1-alpha) + norminv(1-beta)*...
sqrt( p1*(1-p1)/( p0 * (1-p0))))^2/(p0 - p1)^2
% 497.7779 approx 498