About 78% of all female heart transplant patients will survive for at least 3 ye

Question

About 78% of all female heart transplant patients will survive for at least 3 years. Eighty female heart transplant patients are randomly selected. What is the probability that the sample proportion surviving for at least 3 years will be less than 75%? Assume the sampling distribution of sample proportions is a normal distribution. The mean of the sample proportion is equal to the population proportion and the standard deviation is equal to StartRoot StartFraction pq Over n EndFraction EndRoot pq n.

Explanation / Answer

Here, 78% of all female heart transplant patients will survive for at least 3 years.
p = 0.78 and

n = 80

np and n(1-p) both > 10, normal approximation used

standard error of p = sqrt( p(1-p)/n)) = sqrt (0.78 *0.22 /80) =.00214

z value for p=0.75 is z = ( 0.75-0.78)/0.00214 = -0.6478

P( p < 0.70) = P( z < -0.65)
=1P ( Z<0.65 )
=10.7422
=0.2578

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