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 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.