Q1 :- A process is out of control if 4/5observations on the same side of the mean are at least one standarddeviation fr…

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Consider the 5 observations: from a normal probability table,the probability that one observation is >= +1 standard deviationor more from the mean is 0.1587. By the binomialdistribution, the probability that all 5 observations are this faraway or farther is 0.1587^5. Also by the binomialdistribution, the probability that exactly 4 observations are thisfar away or farther is 5*(0.1587^4)*(1-0.1587). These resultscome from the binomial distribution. For example, themultiplier 5 reflects that either the first observation, or thesecond one, or the third one, or the fourth one, or the fifth one,could be the one less than 1 standard deviation above themean. Adding these expressions gives the probability that 4of 5 (or more, namely all 5 -- is the "or more" intended? -- itpresumably is, since the process would then be even moreemphatically out of control) observations are more than one sigmaabove the mean. Note that the use of the binomialdistribution assumes independence of the successive observations --a shaky assumption in industrial practice where a process may bedrifting out of control.

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