Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Help with probability&stat problem
Replies: 47   Last Post: Jun 16, 2007 4:32 AM

 Messages: [ Previous | Next ]
 C6L1V@shaw.ca Posts: 1,532 Registered: 5/23/05
Re: Help with probability&stat problem
Posted: Jun 11, 2007 11:07 PM

On Jun 11, 10:27 am, tutorny <tuto...@gmail.com> wrote:
>

> > 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} = .405854071e-1 = 4.06%,
P_normal{X >= 30} = .282651386e-1 = 2.83%,
P_normal{X >= 29.5} = .369087925e-1 = 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!!

Date Subject Author
6/11/07 tutorny
6/11/07 briggs@encompasserve.org
6/11/07 tutorny
6/11/07 Randy Poe
6/11/07 tutorny
6/11/07 Bruce Weaver
6/11/07 tutorny
6/11/07 Bruce Weaver
6/11/07 C6L1V@shaw.ca
6/11/07 Randy Poe
6/11/07 tutorny
6/11/07 Nick
6/11/07 Jack Tomsky
6/11/07 Luis A. Afonso
6/11/07 Jack Tomsky
6/11/07 Luis A. Afonso
6/11/07 Jack Tomsky
6/12/07 Luis A. Afonso
6/12/07 Jack Tomsky
6/12/07 Luis A. Afonso
6/12/07 Jack Tomsky
6/12/07 Luis A. Afonso
6/13/07 Jack Tomsky
6/13/07 Luis A. Afonso
6/13/07 Anon.
6/13/07 Jack Tomsky
6/13/07 Jack Tomsky
6/13/07 Luis A. Afonso
6/13/07 Jack Tomsky
6/13/07 Anon.
6/13/07 Luis A. Afonso
6/13/07 Jack Tomsky
6/13/07 Jack Tomsky
6/13/07 Anon.
6/13/07 Luis A. Afonso
6/13/07 Jack Tomsky
6/13/07 Luis A. Afonso
6/13/07 Jack Tomsky
6/14/07 Luis A. Afonso
6/14/07 Luis A. Afonso
6/14/07 Jack Tomsky
6/14/07 Luis A. Afonso
6/15/07 Luis A. Afonso
6/15/07 Jack Tomsky
6/15/07 Luis A. Afonso
6/15/07 Jack Tomsky
6/15/07 Luis A. Afonso
6/16/07 Luis A. Afonso