On average 1 in 10 of the chocolates produced in a factory are mis-shapes.In a random sample of 1000 chocolates,find the probability that i) fewer than 80 are mis-shapes.

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Answer 1

The number of unacceptable chocolate bars in a sample of 1000
1000
is a random variable 𝑋
X
which has the binomial distribution with parameters 𝑛=1000
n=1000
and 𝑝=0.10
𝑋∼Bin(𝑛=1000,𝑝=0.10)
i.
To calculate the required probabilities we will use the normal approximation to the binomial distribution, that is
𝑋∼N(𝜇=𝑛𝑝,σ^2=𝑛𝑝(1−𝑝))
X∼N(μ=np,σ^2=np(1−p))
approximately. Here μ=𝑛𝑝=1000⋅0.1=100
μ=np=1000⋅0.1=100
and σ2=𝑛𝑝(1−𝑝)=100⋅0.9=90
σ^2=np(1−p)=100⋅0.9=90
. Applying also the continuity correction we have that

For A)
𝑃(𝑋<80)=𝑃(𝑋≤79)≈𝑃(𝑋−μσ≤79+1/2−10090‾‾‾√)=𝑃(𝑍≤−2.16)==Φ(−2.16)=1−Φ(2.16)=0.015
P(X<80) =P(X≤79)≈P(X−μσ≤79+1/2−10090)
=P(Z≤−2.16)=
Φ(−2.16)=1−Φ(2.16)=0.015
using the normal distribution tables.