3. Suppose we have a binomial experiment with n = 40 trials and a probability of

Question

3. Suppose we have a binomial experiment with n = 40 trials and a probability of success p = 0.85.

a) Is it appropriate to use a normal approximation to this binomial distribution? Why?

b) Compute µ and s of the approximating normal distribution.

c) Use a continuity correction factor to convert the statement r < 30 successes to a statement about the corresponding normal variable x.

d) Estimate P (r<30)

e) Interpretation Is it unusual for a binomial experiment with 40 trials and probability of success 0.85 to have 30 or more successes? Explain  

Explanation / Answer

a)If np and nq both are greater than 5,then we can use normal approximation of the binomial distribution.

Now np=40*0.85=34>5 and nq=40*0.15=6>5

Hence,we can use Normal approximation to this binomail distribution.

b) Mean=np=40*0.85=34

s=standard deviation=sqrt(var)=sqrt(np(1-p))=sqrt(34*0.15)=2.258

c) We use continuity correction factor to improve the approximation.

r<30

30>r

30-0.5>x

29.5>x

x<29.5

d)P(r<30)=P(x<29.5)=P(Z<-1.9929)=P(Z>1.9929)=1-P(Z<1.9929)=1-0.9769=0.0231

( here, z=(29.5-34)/2.258=-1.9929)

e)For binomial experiment with 40 trials and probabilty of success 0.85 to have 30 or more successes,the probabilty of this to happen will be equal to 1-0.0231=0.9769.(P(r>=30)=P(x>=29.5)=P(Z>-1.9929)=P(Z<1.9929)=0.9769

i.e 97.69 % of the time 30 or more successes can happen in binomial experiment with 40 trials and probabilty of success 0.85.

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