Date: Jun 11, 2007 1:02 PM
Subject: Re: Help with probability&stat problem
In article <firstname.lastname@example.org>, tutorny <email@example.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 apparently
read it as the probability that 30 or fewer leave. The longer I look
at 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 = 4.1952
Looks reasonable. And I personally agree that the normal approximation
is a good fit for this kind of question. Especially since we're not
way out on the tail of the curve.
> We want P(x <=30)
> When x = 30, z = (x - u)/s.d = (30 - 22)/4.1952 = 1.9069
Here, I think you've committed a fencepost error. If you're treating
a normal distribution as if it were a discrete histogram then you
want to put your cutoff points between the bars on the histogram, not
in 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 considering
whether the case when x=30 is included or excluded in the set of cases
you are looking for.
Think about it this way. You're approximating p(x=30) in the
discrete 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 probability
that x is 30 or more, do you want the answer to be the complement
of the probability that x is 30 or less? It will be if you use
p(x>=30) and p(x<=30) as your respective estimates. Or do you want the
non-zero probability that x is 30 exactly to figure in somehow?
That's where the 29.5 and 30.5 make themselves useful.