
Re: Help with probability&stat problem
Posted:
Jun 11, 2007 11:07 PM


On Jun 11, 10:27 am, tutorny <tuto...@gmail.com> wrote: > Thanks so much for your reply! > > > 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. > > A lot of us in the class had issue with the wording of the problem. > We emailed the professor, and she said that the correct understanding > is "30 or fewer leave" or P(x <=30), so that's what I've been working > with. > > > 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. > > Exactly the reasoning I used. The book said that normal approximation > should work as long as p isn't too close to 0 or 1, and I figured that > 0.20 should be reasonable for this. > > > > 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. > > I totally missed that point, and looking over the chapter I see that > you are right. Since this is not continuous, I have to use either > 29.5 or 30.5. Since I've confirmed that the question asks for P(x <= > 30) I don't think that I can use 30.5, since that's > 30, so I have to > use 29.5.
Wrong: for the *exact* binomial case we have P{X <= 30} = P{X <= 30.5} (since X can only be an integer, anyway). You should approximate the exact P{X <= 30.5} by its normal value. The exact binomial gives P{X <= 30} = .9752864841 (97.52%), while the normal approximations are P{X <= 30} = .9717348614 (97.17%) and P{X <= 30.5} = .9786231409 (97.86%). Your suggestion to use {X <= 29.5} would give a poorer approximation: P{X <= 29.5} = .9630912075 (96.31%).
Note that if you interpreted the question to mean {X >= 30} instead of {X <= 30}, the normal approximations would NOT be very good. In this case we have P{X >= 30} = P{X >= 29.5} for the exact binomial, so the normal approximation with the "1/2 correction" would be P_normal{X >= 29.5}. Maple 9.5 gives: P_binomial{X >= 30} = .405854071e1 = 4.06%, P_normal{X >= 30} = .282651386e1 = 2.83%, P_normal{X >= 29.5} = .369087925e1 = 3.69%
This should be a warning to you that using the normal approximation is not always good (unless you cannot avoid it), even though you have "large n" and "moderate p".
R.G. Vickson
> > So, for x = 29.5: z = (x  u)/s.d = (29.5  22)/4.1952 = 1.7878 > P(z <= 1.7878) = 0.9631 and the answer is 96.31% which is also > reasonable. > > How is that? > > Thanks!!

