In Europe, 53% flowers of the regardless orchid, Dactylorhiza sambucina, are yel
ID: 3133208 • Letter: I
Question
In Europe, 53% flowers of the regardless orchid, Dactylorhiza sambucina, are yellow, whereas the remaining flowers are purple. For this problem, you may use normal approximation only if it is appropriate to do so.
a. If we took a random sample of a single individual from this population, what is the probability that it would be purple?
b. If we took a random sample of five individuals what is the probability that at least three are yellow?
c. If we took many samples of n=5 individuals, what is the expected stdev of the sampling distribution for the proportion of yellow flowers?
d. if we took a random sample of 263 individuals what is the probability that no more than 150 are yellow?
Explanation / Answer
answer a)
let P(yellow)=p =53%=0.53
therefore P(purple)=1-p=0.47
answer of b)
here we use binomial with n=5, p=0.53
P(x=at least three are yellow)=p(x=0)+p(x=1)+p(x=2)+p(x=3)=0.022935+0.129312+0.291639+0.328869=0.772754
answer of c)
we use binomial distribution properties
here n=5,p=0.53,q=0.47
variance=npq=5*0.53*0.47=1.2455
standard deviantion=sqrt(variance)=1.116
answer of d) here n=263
here mean=263*0.53=139.39
variance=263*.53*.47=65.51
standard deviation=8.09
for P(x>150)=P(z>(150-139.39)/8.09)=P(z>1.3115)=1-P(z<1.3115)=1-.9052=0.0948