Thread: Approximating binomial to normal

Hi All,

Could someone please help with the following.

The probability that a toy balloon coming off a production line is
faulty is 0.02. The balloons are put into bags containing 10
balloons.
(a)Assuming that faulty balloons occur at random, calculate the
probability tht a bag contains a faulty balloon.
The bags are packaged into boxes, each box containing 100 bags.
(b)Using a suitable approximation, estimate the probability that a
box contains 90 or more bags of fault-free balloons.

For part (a) Use a binomial distribution with parameters of n=10 and
p=0.02. Find the probability that X=0 and subtract it from 1 to get
the probability that a bag contains a faulty balloon. This works out
to be 1-0.817 = 0.183.

For (b)Appox. to the Normal with parameter of np=100 x 0.817=81.7
and npq=100 x 0.817 x 0.183 =14.94
P(X>=90)=P(V>=89.5) (considering continuity correction)
Let Z=V-81.7/(sqr-root(14.94))
P(Z>=(89.5-81.7)/(sqr-root(14.94))
=P(Z>=2.0173)
=1 - P(Z<2.0173)
=1-0.9781
=0.0219
However, the book is giving an answer of 0.013. Could someone please
tell me where I have made a mistake??
Thankyou.

Originally Posted by woollybull

Hi All,

Could someone please help with the following.

The probability that a toy balloon coming off a production line is
faulty is 0.02. The balloons are put into bags containing 10
balloons.
(a)Assuming that faulty balloons occur at random, calculate the
probability tht a bag contains a faulty balloon.
The bags are packaged into boxes, each box containing 100 bags.
(b)Using a suitable approximation, estimate the probability that a
box contains 90 or more bags of fault-free balloons.

For part (a) Use a binomial distribution with parameters of n=10 and
p=0.02. Find the probability that X=0 and subtract it from 1 to get
the probability that a bag contains a faulty balloon. This works out
to be 1-0.817 = 0.183.

For (b)Appox. to the Normal with parameter of np=100 x 0.817=81.7
and npq=100 x 0.817 x 0.183 =14.94
P(X>=90)=P(V>=89.5) (considering continuity correction)
Let Z=V-81.7/(sqr-root(14.94))
P(Z>=(89.5-81.7)/(sqr-root(14.94))
=P(Z>=2.0173)
=1 - P(Z<2.0173)
=1-0.9781
=0.0219
However, the book is giving an answer of 0.013. Could someone please
tell me where I have made a mistake??
Thankyou.

It looks like the book hasn't used the continuity correction .......