Question

A drug test is accurate 91% of the time. If the test is given to 225 people who have not taken drugs, what is the probability that exactly 201 will test negative?

First, use the normal approximation to the binomial, and use the continuity correction. Normal Approximate Probability = ?

Now use the binomial distribution.

Exact Binomial Probability = ?

Does the normal distribution make a pretty good approximation to the binomial, in this case?

Yes, that looks pretty good. It must be because n*p >= 5 AND n*(1-p) >= 5.

or

No, those probabilities are far apart. It must be because n*p < 5, or else n*(1-p) < 5

Answer 1

1)

normal approximation :

here mean of distribution=μ=np=		204.75
and standard deviation σ=sqrt(np(1-p))=		4.29
for normal distribution z score =(X-μ)/σx

therefore from normal approximation of binomial distribution and continuity correction:

probability =P(200.5<X<201.5)=P((200.5-204.75)/4.293)<Z<(201.5-204.75)/4.293)=P(-0.99<Z<-0.76)=0.2236-0.1611=0.0625

2)

binomial distribution:

P(X=201) =binomdist(201,225,0.91,false)=0.0598

3)

Yes, that looks pretty good. It must be because n*p >= 5 AND n*(1-p) >= 5.