normal distribution?

[匿名]

2015-12-05 6:33 am

A factory makes bottles of juice using only two machines,A and B.The volume of the juice produced by them are normally distributed.
Machine A produces bottles of juice with a mean volume of 500ml, and a variance of 1.96ml^2,while machine B produces bottles with a mean volume of 498ml,and a variance of 4ml^2.It is also known that 0.4 of the output of the factory has a volume less than 499ml.
(a)Find the probability that a randomly selected bottle is produced by machine A.
(b)If a bottle of juice produced by this factory is found to have a volume less than 499ml,what is the probability that it is produced by machine A?

回答 (1)

Lopez

2015-12-05 12:02 pm

✔ 最佳答案

(a)
s(A) = √1.96 = 1.4
s(B) = √4 = 2

Suppose p = P( bottle by machine A )

p * P( Va < 499 ) + (1-p) * P( Vb < 499 ) = 0.4
p * P( Z < (499-500)/1.4 ) + (1-p) * P( Z < (499-498)/2 ) = 0.4
p * P( Z < - 0.71 ) + (1-p) * P( Z < 0.5 ) = 0.4
p*0.2389 + (1-p)*0.6915 = 0.4
p = (0.6915-0.4)/(0.6915-0.2389) ≒ 0.644

(b)
P( bottle by machine A ∣ V < 499 )
= P( bottle by machine A and V < 499 ) / P( V < 499 )
= p * P( Va < 499 ) / P( V < 499 )
= 0.644*0.2389 / 0.4
≒ 0.385

Ans:
(a) 0.644
(b) 0.385

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