guessing mean of bounded variable with 1:30 sampling ratio

[nicola.baldini2]

I have a population of N=12000. I want to know the mean (and possibly the standard deviation) of a variable x, bounded between 1 and 7. I took a (let's suppose random) sample of n=400 and estimated mean = 3.14 (standard error = .15) and standard deviation = 2.28. The sample strongly departs from normality. Can I trust such estimates? How much? Can I attach a p-value to them? I need to state formally that, despite a ridicolous response rate, my research is not that bad.
Nicola

[statisticsdoc]

Nicola,

You have the Central Limit Theorem working for you here.  Even though the
distribution of individual cases is not normal, the distribution of sample
means (with a sample size of 400) will approximate the normal distribution
and should provide you with a reasonable estimate of the population mean and
the standard error of the means of samples of 400 cases.

HTH,

Stephen Brand

For personalized and professional consultation in statistics and research
design, visit
www.statisticsdoc.com


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]]On Behalf Of
Nicola Baldini
Sent: Friday, December 08, 2006 6:11 AM
To: [hidden email]
Subject: guessing mean of bounded variable with 1:30 sampling ratio


I have a population of N=12000. I want to know the mean (and possibly the
standard deviation) of a variable x, bounded between 1 and 7. I took a
(let's suppose random) sample of n=400 and estimated mean = 3.14 (standard
error = .15) and standard deviation = 2.28. The sample strongly departs from
normality. Can I trust such estimates? How much? Can I attach a p-value to
them? I need to state formally that, despite a ridicolous response rate, my
research is not that bad.
Nicola

[Richard Ristow]

At 06:11 AM 12/8/2006, Nicola Baldini asked:

>I have a population of N=12000. I want to know the mean (and possibly
>the standard deviation) of a variable x, bounded between 1 and 7. I
>took a (let's suppose random) sample of n=400 and estimated mean =
>3.14 (standard error = .15) and standard deviation = 2.28. Can I trust
>such estimates?

To which, at 10:41 AM 12/9/2006, Stephen Brand replied:

>You have the Central Limit Theorem working for you here.  Even though
>the
>distribution of individual cases is not normal, the distribution of
>sample
>means (with a sample size of 400) will approximate the normal
>distribution
>and should provide you with a reasonable estimate of the population
>mean and the standard error of the means of samples of 400 cases.

To which I'll add, the Central Limit Theorem has an important ally
here. Because your population mean and standard deviation are bounded
(1<=mean<7; 0<=SD<=2.5, if my arithmetic's right*), convergence should
be rapid, plenty good enough with n=400. THAT's not your problem.

Here's your problem: "I took a (let's suppose random) sample of n=400."
Nope; no supposing. The arguments using the Law of Large Numbers and
Central Limit Theorem only apply if the sample is random. You need to
have a decent argument that you have a random sample, or at least that
your sampling distribution is independent of the variable x.

You wave a big red flag:  "I need to state formally that, despite a
ridiculous response rate, my research is not that bad." 'Response
rate': your 400 are respondents to a survey? How many did you survey -
all 12,000? If so, there's no practical chance that a 3% response rate
is a random sample of the population, not even approximately.

If you sampled a fraction, selected randomly, and had a higher response
rate within that fraction, you may have a good argument. Otherwise, I'm
afraid not likely.

[statisticsdoc]

Nicola,

Richard raises an important point about random sampling.  Did you mean to
say that the response rate was 400/12000 (as stated  in the text of your
message), or did you mean that the sampling ration was 400/12000 (i.e.,
n/N).  If the former, random sampling seems highly unlikely.  If the latter,
random sampling is possible but should be documented (e.g., how were the
samples drawn?  Did the samples resemble the population in terms of other
descriptive characteristics?).

HTH,

Stephen Brand


For personalized and professional consultation in statistics and research
design, visit
www.statisticsdoc.com


-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]]On Behalf Of
Nicola Baldini
Sent: Friday, December 08, 2006 6:11 AM
To: [hidden email]
Subject: guessing mean of bounded variable with 1:30 sampling ratio


I have a population of N=12000. I want to know the mean (and possibly the
standard deviation) of a variable x, bounded between 1 and 7. I took a
(let's suppose random) sample of n=400 and estimated mean = 3.14 (standard
error = .15) and standard deviation = 2.28. The sample strongly departs from
normality. Can I trust such estimates? How much? Can I attach a p-value to
them? I need to state formally that, despite a ridicolous response rate, my
research is not that bad.
Nicola