Finding the probability using the central limit theorem

Question

QUESTION:

 

Assume 37% of adults take their favorite color is blue. In a sample of 36 randomly selected adults, find the estimated probability that more than 20 adults take their favorite color is blue.

 

So here's what I know:

n=36

p=.37

q=.63

Mu=13.32

Sigma=2.90

x= the number of adults that say their favorite color is blue out of 36

 

My question is what do I do from here? I am not sure how to set up the problem and what formulas to use when and where. Thank you in advance!

Answer 1

You are interested in P(x>20) but using the central limit theorem or normal distribution approximation. To do so, we do the following,

P(x > 20) = P((x-np)/(npq)^0.5 > (20-np)/(npq)^0.5)  ≅ P(z> 2.305972) = 0.011

where  ≅ is the approximation due to the central limit theorem (because both np and nq are greater than 5).

 

We can improve the above result by including a continuity correction for using a normality distribution approximation to find a probability involving a discrete distribution.  

 

I hope this helps.

 

Thanks,

Kidane