Suppose a simple random sample of size nequals=75 is obtained from a population

Question

Suppose a simple random sample of size nequals=75 is obtained from a population whose size is N=20,000 and whose population proportion with a specified characteristic is p=0.6. (a) Describe the sampling distribution. Choose the phrase that best describes the shape of the sampling distribution below. A. Approximately normal because n0.05N and np(1p)<10. B.Not normal because n0.05N and np(1p)<10. C.Not normal because n0.05N and np(1p)10. D.Approximately normal because n0.05N and np(1p)10. (b) What is the probability of obtaining x=51 or more individuals with the characteristic? That is, what is P(0.68)? (c) What is the probability of obtaining xs=42 or fewer individuals with the characteristic? That is, what is P(0.56)?

Explanation / Answer

Approximately normal because n0.05N and np(1p)10.

b)mean =np=75*0.6=45

st deviation =(np(1-p))1/2 =4.24

P(X=>51) =1-P(X<=50)=1-P(Z<(50.5-45)/4.25)=1-P(Z<1.2964)=1-0.9026=0.0974

P(X<=42)=P(Z<(42.5-45)/4.25)=P(Z<-0.5893)=0.2778

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