The amount of fill in a half-liter (500 ml) soft drink bottle is normally distri

Question

The amount of fill in a half-liter (500 ml) soft drink bottle is normally distributed. The process has a standard deviation of 3.5 ml. The mean is adjustable.

(a) Where should the mean be set to ensure a 90 percent probability that a half-liter bottle will not be underfilled? (Round your answer to 2 decimal places.)  

Mean             ml

(b) Where should the mean be set to ensure a 95 percent probability that a half-liter bottle will not be underfilled? (Round your answer to 2 decimal places.)  

Mean             ml

(c) Where should the mean be set to ensure a 99.0 percent probability that a half-liter bottle will not be underfilled? (Round your answer to 2 decimal places.)  

Mean             ml

rev: 02_26_2016_QC_CS-43403

References

Explanation / Answer

Solution:

Let X denotes the amount of fill in half-liter soft drink bottle and follows a normal distribution wih paramaters (, = 3.5 ml)
a) Find the mean be set to ensure a 90% of probability is,
P(X 500) = 90%
1- P(Z < 500-/3.5) = 0.90
P(Z < 500-/3.5) = 1-0.90
500-/3.5 = 0.10
500-/3.5 = 1.282
500- = 4.48700
= 500-4.48700
= 495.513
Hence, the required mean is 495.513

b) Find the mean be set to ensure a 95% of probability is,
P(X 500) = 95%
1- P(Z < 500-/3.5) = 0.95
P(Z < 500-/3.5) = 1-0.95
500-/3.5 = 0.05
500-/3.5 = -1.645
500- = -5.7575
= 500-5.7575
= 494.243
Hence, the required mean is 494.243

c) Find the mean be set to ensure a 99% of probability is,
P(X 500) = 99%
1- P(Z < 500-/3.5) = 0.99
P(Z < 500-/3.5) = 1-0.99
500-/3.5 = 0.01
500-/3.5 = -2.33
500- = -8.15500
= 500-8.15500
= 491.845
Hence, the required mean is 491.845

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