Question 1010221: For the normally distributed random variables, find the probabilities
1) P (x-bar > 2.5), if μ = 2.3, σ = 0.8, n = 50
2) P (x-bar < 3.1), if μ = 3.0, σ = 1.0, n = 100
Answer by mathmate(429)   (Show Source):
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Question:
Find probabilities of the following normally distributed random variables:
1) P (x-bar > 2.5), given μ = 2.3, σ = 0.8, n = 50
2) P (x-bar < 3.1), given μ = 3.0, σ = 1.0, n = 100
Solution:
Standard deviation of sample mean = σ/sqrt(n)
Z=(x-μ)/σ
1. S.D. of sample mean = 0.8/sqrt(50) = 0.11314
P((x-bar) > 2.5)
=P(Z>(2.5-2.3)/0.13114)
=P(Z>1.7678)
=1-P(Z<1.7678)
=1-0.96145
=0.03855
2. S.D. of sample mean = 1.0/sqrt(100)=0.10
P((x-bar)<3.1)
=P(Z<(3.1-3.0)/0.10)
=P(Z<1)
=0.84134
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