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Topic: [ap-stat] 30 and the Central Limit Theorem
Replies: 6   Last Post: Mar 23, 2012 7:21 PM

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Bob

Posts: 1,360
Registered: 6/10/11
Re:[ap-stat] 30 and the Central Limit Theorem
Posted: Mar 22, 2012 3:14 PM
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Forwarded message:
> From: Jared Derksen <mrmathman@gmail.com>
>
> Chris:
>
> This is helpful. And interesting! As usual, your reference knowledge is
> impressive.
>
> So does a more nuanced explanation go like this?
>
> There are some mathematical and historical reasons to believe that when n >
> 30, this is a safe point at which to believe that a sample mean is becoming
> approximately normal.


It's certainly historical -- there isn't much mathematical support;-)
I suspect it was a confusion between how big n has to be for z to
approximate t with how big n has to be for the sampling distribution
to be approximately normal. It might also be that you need at least
n=30 before looking at the sample data gives you a clue as to what is
happening in the population. (Bassed on recent simulations, i'd say
that is too optimistic.)

In any case, at the risk of appearing to agree with Chris, I suggest
the following compromise:

IF n>30 AND we see no trouble ion the plot, we are probably safe.

> However, we still like to graph our data, when possible, as an indication
> of how skewed the population might be. And how confident we are that the
> mean we are working with has an approximately normal distribution.
>
> And the practice switching from t to z at 30 (or 40) is an artifact of a
> lack of inadequate technology. Now that we can easily compute t for any
> sample size, there is no need act as if something magically changes at 30.


Is there one too many negatives here? Anyway, as I said earlier,
adequate tables have been around for as long as significant numbers of
people were using t. I think the tables are a red herring.

> Is that more complete?
>
> Cheers!
>
> JD
>
> On Thu, Mar 22, 2012 at 10:05 AM, Chris Olsen <COlsen@mchsi.com> wrote:
>

> > Jared and All =96****
> >
> > ** **
> >
> > The =9330 is enough=94 goes all the way back to a very early publicati=

> on by
> > Student (Wm Gossett). In an article somewhere in the vicinity of 1922 or
> > so, he says something like, =93=85and with sample sizes are about 30, one=

> can
> > use the normal curve.=94****
> >
> > ** **
> >
> > Another Wm, Wm Cochrane, has an article that pretty much justifies the
> > number 30.****
> >
> > ** **
> >
> > Sorry, I don=92t have either reference at hand, and I=92m kinda presse=

> d for
> > time. But if anyone would like the refs, send me an e-mail and I=92ll =
> try
> > to hit the library or search my office back at Grinnell. We=92re on brea=
> k,
> > so don=92t hold your breath=85****
> >
> > ** **
> >
> > **n **Chris ****
> >
> > ** **
> >
> > *From:* Jared Derksen [mailto:mrmathman@gmail.com]
> > *Sent:* Thursday, March 22, 2012 9:57 AM
> > *To:* AP Statistics
> > *Cc:* AP Statistics
> > *Subject:* Re: [ap-stat] Question regarding normality check for a two
> > sample t-test****
> >
> > ** **
> >
> > 30 comes from 30 lines of text on a t-table. Books with smaller font use=

> d
> > 40...****
> >
> > ** **
> >

>
>
>
> --=20
> Jared Derksen
>
> I understand the need for conformity. Without a concise set of rules to
> follow we would probably all have to resort to common sense. --David
> Thorne
>
> Frequently asked questions(FAQ) http://web.me.com/mrmathman/MrMathMan/FAQ.h=
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>
> Chris:<div><br></div><div>This is helpful. =A0And interesting! =A0As usual,=
> your reference knowledge is impressive.</div><div><br></div><div>So does a=
> more nuanced explanation go like this?</div><div><br></div><div>There are =
> some mathematical and historical reasons to believe that when n &gt; 30, th=
> is is a safe point at which to believe that a sample mean is becoming appro=
> ximately normal. =A0</div>
> <div><br></div><div>However, we still like to graph our data, when possible=
> , as an indication of how skewed the population might be. =A0And how confid=
> ent we are that the mean we are working with has an approximately normal di=
> stribution.</div>
> <div><br></div><div>And the practice switching from t to z at 30 (or 40) is=
> an artifact of a lack of inadequate technology. =A0Now that we can easily =
> compute t for any sample size, there is no need act as if something magical=
> ly changes at 30.</div>
> <div><br></div><div>Is that more complete? =A0</div><div><br></div><div>Che=
> ers!</div><div><br></div><div>JD<br><br><div class=3D"gmail_quote">On Thu, =
> Mar 22, 2012 at 10:05 AM, Chris Olsen <span dir=3D"ltr">&lt;<a href=3D"mail=
> to:COlsen@mchsi.com">COlsen@mchsi.com</a>&gt;</span> wrote:<br>
> <blockquote class=3D"gmail_quote" style=3D"margin:0 0 0 .8ex;border-left:1p=
> x #ccc solid;padding-left:1ex"><div lang=3D"EN-US" link=3D"blue" vlink=3D"p=
> urple"><div><p class=3D"MsoNormal"><span style=3D"font-size:11.0pt;font-fam=
> ily:&quot;Calibri&quot;,&quot;sans-serif&quot;;color:#1f497d">Jared and All=
> =96<u></u><u></u></span></p>
> <p class=3D"MsoNormal"><span style=3D"font-size:11.0pt;font-family:&quot;Ca=
> libri&quot;,&quot;sans-serif&quot;;color:#1f497d"><u></u>=A0<u></u></span><=
> /p><p class=3D"MsoNormal"><span style=3D"font-size:11.0pt;font-family:&quot=
> ;Calibri&quot;,&quot;sans-serif&quot;;color:#1f497d">=A0=A0 The =9330 is en=
> ough=94 goes all the way back to a very early publication by Student (Wm Go=
> ssett).=A0 In an article somewhere in the vicinity of 1922 or so, he says s=
> omething like, =93=85and with sample sizes are about 30, one can use the no=
> rmal curve.=94<u></u><u></u></span></p>
> <p class=3D"MsoNormal"><span style=3D"font-size:11.0pt;font-family:&quot;Ca=
> libri&quot;,&quot;sans-serif&quot;;color:#1f497d"><u></u>=A0<u></u></span><=
> /p><p class=3D"MsoNormal"><span style=3D"font-size:11.0pt;font-family:&quot=
> ;Calibri&quot;,&quot;sans-serif&quot;;color:#1f497d">=A0=A0 Another Wm, Wm =
> Cochrane, has an article that pretty much justifies the number 30.<u></u><u=

> ></u></span></p>
> <p class=3D"MsoNormal"><span style=3D"font-size:11.0pt;font-family:&quot;Ca=
> libri&quot;,&quot;sans-serif&quot;;color:#1f497d"><u></u>=A0<u></u></span><=
> /p><p class=3D"MsoNormal"><span style=3D"font-size:11.0pt;font-family:&quot=
> ;Calibri&quot;,&quot;sans-serif&quot;;color:#1f497d">=A0=A0 Sorry, I don=92=
> t have either reference at hand, and I=92m kinda pressed for time.=A0=A0=A0=
> But if anyone would like the refs, send me an e-mail and I=92ll try to hit=
> the library or search my office back at Grinnell.=A0 We=92re on break, so =
> don=92t hold your breath=85<u></u><u></u></span></p>
> <p class=3D"MsoNormal"><span style=3D"font-size:11.0pt;font-family:&quot;Ca=
> libri&quot;,&quot;sans-serif&quot;;color:#1f497d"><u></u>=A0<u></u></span><=
> /p><p style=3D"margin-left:24.75pt"><u></u><span style=3D"font-size:11.0pt;=
> font-family:Wingdings;color:#1f497d"><span>n<span style=3D"font:7.0pt &quot=
> ;Times New Roman&quot;">=A0 </span></span></span><u></u><span style=3D"font=
> -size:11.0pt;font-family:&quot;Calibri&quot;,&quot;sans-serif&quot;;color:#=
> 1f497d">Chris <u></u><u></u></span></p>
> <p class=3D"MsoNormal"><span style=3D"font-size:11.0pt;font-family:&quot;Ca=
> libri&quot;,&quot;sans-serif&quot;;color:#1f497d"><u></u>=A0<u></u></span><=
> /p><p class=3D"MsoNormal"><b><span style=3D"font-size:10.0pt;font-family:&q=
> uot;Tahoma&quot;,&quot;sans-serif&quot;">From:</span></b><span style=3D"fon=
> t-size:10.0pt;font-family:&quot;Tahoma&quot;,&quot;sans-serif&quot;"> Jared=
> Derksen [mailto:<a href=3D"mailto:mrmathman@gmail.com" target=3D"_blank">m=
> rmathman@gmail.com</a>] <br>
> <b>Sent:</b> Thursday, March 22, 2012 9:57 AM<br><b>To:</b> AP Statistics<b=
> r><b>Cc:</b> AP Statistics<br><b>Subject:</b> Re: [ap-stat] Question regard=
> ing normality check for a two sample t-test<u></u><u></u></span></p><p clas=
> s=3D"MsoNormal">
> <u></u>=A0<u></u></p><p class=3D"MsoNormal">30 comes from 30 lines of text =
> on a t-table. =A0Books with smaller font used 40...<u></u><u></u></p><div><=
> p class=3D"MsoNormal"><u></u>=A0<u></u></p></div></div></div></blockquote><=
> /div><br>
> <br clear=3D"all"><div><br></div>-- <br>Jared Derksen<br><br><span style=3D=
> "color:rgb(95,103,109);font-family:Arial,Helvetica,sans-serif;font-size:11p=
> x;line-height:15px">I understand the need for conformity. Without a concise=
> set of rules to follow we would probably all have to resort to common sens=
> e. =A0 --David Thorne<br>
> </span><br>
> </div>
>
> <p>Frequently asked questions (FAQ):&#160;<a href=3D"http://web.me.com/mrma=
> thman/MrMathMan/FAQ.html">http://web.me.com/mrmathman/MrMathMan/FAQ.html</a=

> ><br />
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