Let X be a binomial rv based on n trials with success probability p. Then if the
binomial probability histogram is not too skewed, X has approximately a
normal distribution with μ=np and σ=npq.
In practice, the approximation is adequate provided that both np≥10 and nq≥10.
Given n=100,p=0.3,q=1−p=1−0.3=0.7.
np=100(0.3)=30≥10
nq=100(0.7)=70≥10
We can use normal approximation for binomial distribution with μ=np=100(0.3)=30,σ=npq=100(0.3)(0.7)=21
Let Xˉ= the sample mean: Xˉ∼N(μ,σ2/n1)
Given n1=50.
a)
P(Xˉ>3.9)=1−P(Z≤21/503.9−30)
≈1−P(Z≤−40.2732)≈1
b)
P(4.1<Xˉ<4.4)=P(Z<21/504.4−30)
−P(Z≤21/504.1−30)
≈P(Z<−39.5017)−P(Z≤−39.9646)≈0
c)
P(Xˉ<4.0)=P(Z<21/504.0−30)
≈P(Z<−40.1189)≈0
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