About78of all female heart transplant patients will survive

About78%of all female heart transplant patients will survive for at least 3 years. Ninety 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 70%?

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 EndRootpqn. The probability that the sample proportion surviving for at least 3 years will be less than 70% is?

(Round to four decimal places as needed.)

Solution

Here, the standard error of the population proportion is

sp = sqrt (p (1 - p) / n ) = sqrt(0.78*(1-0.78)/90) = 0.043665394

We first get the z score for the critical value. As

z = (p - po) / sp, then as          
          
po = critical value =    0.7      
p = mean =    0.78      
          
sp = standard deviation =    0.043665394      
          
Thus,          
          
z = (p- po) / sp =    -1.832114466      
          
Thus, using a table/technology, the left tailed area of this is          
          
P(z >   -1.832114466   ) =    0.033467184 [ANSWER]