Question
please explain in detail thank you
You work for a pharmaceuticals company as a statistical process analyst Your ob is to analyze processes and make sure they are in statistical control, In one process, a machine is supposed to add vial. (Assume process can be imated by a normal distribution The accept approxi is 965 range of amounts of the compound added milligrams, inclusive valve, the setting Because of an error with the release check machine mis. To shifts" from 98 you select adding the correct amount of the compound into the vials, at random three samples of five vials and find the mean amount of the compound added for each sample. coworker asks why you take of sizes and the mean instead of measuring the amounts in 15 vials individually to check the machine's settings. (Nore. Both samples are chosen without EXERCISES 1. Sampling Individuals You select one vial and determine how much of the compound was added. Assume the machine shifts and the distribution of acceptable he compound added now has a mean amount 9.96 milligrams and a standard deviation of 0.05 milligram that you select a vial that (a) What outside the acceptable range (in other words, you do not detect that the machine has shifted)? (See figure.) (b) You randomly select 15 vials what al that bability FIGURE FOR EXERCISE 1 acceptable range? 2. sampling Groups of Five original You select five vials and find the mean amount of compound added. Assume the machine shifts and is filling the vials with a mean amount of 996 milligrams and a standard deviation of milligram. (a) What is the probability that you select a sample of five vials that has a mean that is not outside the acceptable range? (b) You randomly select three samples of five vials What is the probability that you select at least one sample of five vials that FIGURE FOR EXERCISE 2 has a mean that is not outside the acceptable range? (c) What is more seasitive to change-an individual measure 3. Writing an Explanation a paragraph to your coworker explaining why you take of of and find the mean of each sample instead randomly choosing and the amounts individually to check the machine's setting 292 APTER s NORMAL PROaABILITY DISTRIBuTIONs
Explanation / Answer
1a) The transformation is given by z = (x - mu)/sigma where mu = 9.96 and sigma = .05. The acceptable range for process is 9.8 to 9.95. So, the prob that vial selected in not ouside range is
p = P(9.65 < X <9.95) = P(-6.2 < Z < -.2) = .42
b) Prob that all 15 vials are outside range = (1- p)15 = .0003
Hence prob that atleast one is within range = 1- (1- p)15 = .9997
2a) The sample of n = 5 vials is distributed Normally. X ~ N(mu, sigma/sqrt(n))
Corresponding z transformation is (x - mu)/sigma/sqrt(n)
So, the prob that sample mean is not ouside range is
p = P(9.65 < X <9.95) = P(-6.2*sqrt(5) < Z < -.2*sqrt(5))
= P(-13.86 < Z < -.45) = .33
b) Probability that all 3 samples (of 5 vials each) fall outside range = (1-.33)3 = .3
So, prob of atleast one sample in range = 1-.3 = .7
c) A mean is more sensitive to change as compared to a single measure
3) The goal of any statistical process control is to identify outliers or samples that cause process to fall out of control. A single sample of size 15 has mean = m and std dev = sigma whereas 3 samples of 5 each has std dev sigma/sqrt(5). A lower value of deviation implies narrower acceptable region. This ensures that process is more accurate.
