Question

A drug test is accurate 95% of the time. If the test is given to 200 people who have not taken drugs, what is the probability that exactly 191 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?

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

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

Answer 1

1)

for normal approximation:

here mean of distribution=μ=np=		190.00
and standard deviation σ=sqrt(np(1-p))=		3.08
for normal distribution z score =(X-μ)/σx

therefore from normal approximation of binomial distribution and continuity correction:

probability =P(190.5<X<191.5)=P((190.5-190)/3.082)<Z<(191.5-190)/3.082)=P(0.16<Z<0.49)=0.6879-0.5636=0.1243

2)

from Binomial probability=binomdist(191,200,0.95,false)=0.1277

3)

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