```Date: Jun 11, 2007 1:02 PM
Author: briggs@encompasserve.org
Subject: Re: Help with probability&stat problem

In article <1181579232.176149.92070@c77g2000hse.googlegroups.com>,  tutorny <tutorny@gmail.com> writes:> I am looking at solving the following problem:> > Past records show that at a given college 20% of the students who> began as psychology majors either changed their major or dropped out> the school.  An incoming class has 110> beginning psychology majors.  What is the probability that as many as> 30 of these students leave the psychology program?I read that as the probability that 30 or more leave.  You've apparentlyread it as the probability that 30 or fewer leave.  The longer I lookat the question, the less sure I am which of us is correct.> I think that I can solve it using the normal approximation to the> binomial probability distribution, as follows:> > n =110, p = 0.20> mean = u = np = 110*0.20 = 22> standard deviation = s.d. = (n*p*q)^.5 = (110*.20*.80)^.5 = 4.1952Looks reasonable.  And I personally agree that the normal approximationis a good fit for this kind of question.  Especially since we're notway out on the tail of the curve.> We want P(x <=30)> > When x = 30, z = (x - u)/s.d = (30 - 22)/4.1952 = 1.9069Here, I think you've committed a fencepost error.  If you're treatinga normal distribution as if it were a discrete histogram then youwant to put your cutoff points between the bars on the histogram, notin the middle of the bars.  You want to look at x=29.5 or x=30.5.You decide whether to use the x=29.5 or the x=30.5 cutoff by consideringwhether the case when x=30 is included or excluded in the set of casesyou are looking for.Think about it this way.  You're approximating p(x=30) in thediscrete case by p(x<=30.5) - p(x<=29.5) in the continuous model.Or think about it this way.  If you were asked for the probabilitythat x is 30 or more, do you want the answer to be the complementof the probability that x is 30 or less?  It will be if you usep(x>=30) and p(x<=30) as your respective estimates.  Or do you want thenon-zero probability that x is 30 exactly to figure in somehow?That's where the 29.5 and 30.5 make themselves useful.
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