Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

It is known that 75% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 60 new products, find the following probabilities. (Round your answers to four decimal places.)

(a) within 2 years 47 or more fail


(b) within 2 years 58 or fewer fail


(c) within 2 years 15 or more succeed


(d) within 2 years fewer than 10 succeed

Answer 1

here for binomial distribution parameter n=60 and p=0.75

here mean of distribution=μ=np=		45.0000
and standard deviation σ=sqrt(np(1-p))=		3.3541
for normal distribution z score =(X-μ)/σx
since np and n(1-p) both are greater than 5, we can use normal approximation of binomial distribution
therefore from normal approximation of binomial distribution and continuity correction:

a)

probability =P(X>46.5)=P(Z>(46.5-45)/3.354)=P(Z>0.45)=1-P(Z<0.45)=1-0.6736=0.3264

b)

probability =P(X<58.5)=(Z<(58.5-45)/3.354)=P(Z<4.02)=1.0000

c)

P( 15 or more succeed ) =P(at most 45 fail):

probability =P(X<45.5)=(Z<(45.5-45)/3.354)=P(Z<0.15)=0.5596

d)

P( fewer than 10 succeed ) =P(at least 51 succeed):

probability =P(X>50.5)=P(Z>(50.5-45)/3.354)=P(Z>1.64)=1-P(Z<1.64)=1-0.9495=0.0505