Question

You work for a pharmaceuticals company as a statistical process analyst. Your job is to analyze processes and make sure they are in statistical control. In one process, a machine is supposed to add 9.8 milligrams of a compound to a mixture in a vial. (Assume this process can be approximated by a normal distribution with a standard deviation of 0.05.) The acceptable range amounts of the compound added is 9.65 milligrams to 9.95 milligrams, inclusive. Because of an error with the release valve, the setting on the machine "shifts" from 9.8 milligrams. To check that the machine is adding the correct amount of the compound into the vials, you select at random three samples of five vials and find the mean amount of the compound added to each sample. A coworker asks why you take 3 samples of size 5 and find the mean instead of randomly choosing and measuring the amounts in 15 vials individually to check the machine's settings. (Note: Both samples are chosen without replacement.) Question: Assume the machine shifts and the distribution of the amount of the compound asses now has a mean of 9.96 milligrams and a standard deviation of 0.05 milligram. You select one vial and determine how much of the compound was added.

Answer 1

(This is the probability of selecting at least 1 vial from the sample of 15 vials, which will be within the acceptable range.)

to 9.98 Now, the X = and cent of the Compound asses, X~ Normal (M=9.96 = 0.05). is 9.65 acceptable range mg mg Hence, pros, (within acceptable renge) = P(9.65 EX 29.95) plx 49,95) - P(x € 9.65) - P(Z €9.98=4)- RD P(Z <-0, 2) - P(24-0.63) P(2 € 9.65 0.4207 p (say? Nord, Ya ol vials in the sample of is, which within acceptable acceptable range . ~ Binoncial (n=15, pa 0,4 207). 1 - ply=0) I- þ° (i-) 's I- P(YLD To find, Ply 7, 1) (15) 0.9997