In the following problem, check that it is appropriate to use the normal approxi

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 79% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 69 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

Explanation / Answer

here for np =69*0.79=54.51 and n(1-p) =14.49 both are greater then 10 ; therefore  it is appropriate to use the normal approximation.

a)

here mean =np =69*0.79=54.51

std deviation =(np(1-p))1/2 =3.38

a)hence probability 47 or more fail =P(X>=47)=1-P(X<=46) =1-P(Z<(46.5-54.51)/3.38)=1-P(Z<-2.3675)

=1-0.0090 =0.9910
b)  58 or fewer fail =P(X<=58)=P(Z<(58.5-54.51)/3.38)=P(Z<1.1793)=0.8809

c)15 or more succeed=P(at most 54 fail) =P(X<=54)=P(Z<(54.5-54.51)/3.38=P(Z<-0.0030)=0.4988

d) P( fewer then 10 succeed )=P( 60 or more failed) =P(X>=60)=1-P(X<=59)=1-P(Z<(59.5-54.51)/3.38)

=1-P(Z<1.4749)=1-0.9299 =0.0701

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