Sampling Error

[CFAZod]

Bear with me, I have no formal statistics education, and I recognize that this point is insignificant considering the actual Level 1 examination. Any difference between the sample mean and the population mean is called the sampling error. In the text, we are told that it is not often possible to calculate population parameters from all observed values. Sample statistics are used to estimate unknown population parameters. Isn't there a paradox somwhere here? We sample to determine unknown population parameters, there may be bias, but the population parameter is not known?

 

[mrnome201]

i wouldn't call it a paradox....
we take samples because in many cases it is impossible to include an entire population.
with samples, you are subject to Type I and Type II errors that result in certain confidence intervals.

 

[JoeyDVivre]

CFAZod Wrote:
-------------------------------------------------------
> Bear with me, I have no formal statistics
> education, and I recognize that this point is
> insignificant considering the actual Level 1
> examination. Any difference between the sample
> mean and the population mean is called the
> sampling error. In the text, we are told that it
> is not often possible to calculate population
> parameters from all observed values. Sample
> statistics are used to estimate unknown population
> parameters. Isn't there a paradox somwhere here?

What's the paradox?

> We sample to determine unknown population
> parameters,

We sample to estimate population parameters.

>there may be bias,

Not in the case of the sample mean which is unbiased for the population mean.

> but the population
> parameter is not known?

Right, the population parameter is either something that we would need an entire census for (e.g. average height of the US population) or is an unknowable property of the process (e.g., expected amount a stock moves per day).

 

[CFAZod]

The paradox (not paradox), or contradiction, occurs by defining sampling error. My shortcoming is in not understanding how to calculate the difference between the estimated statistic and the unknown in order to determine the error or bias. I suppose because a sample statistic may have deviation, there must be a difference between it and the unknown parameter, hence sampling error. Hence confidence intervals to be more certain about an uncertain quantity.

 

[CFAZod]

Let me quote two sentences from the text, page 443, Reading 11:

"The sample mean will provide the analyst with an estimate of the population mean expenditure(for instance from previous sentences involving mean equipment expenditures). Any difference between the sample mean and the population mean is called sampling error."

If in any particular case the population mean cannot be determined (mean expected cash flow returns on outstanding bonds), how then can the difference between the sample mean and the population mean be calculated or even estimated?

 

[JoeyDVivre]

Ain't it great?

So we don't know the population mean (mu) and maybe we don't even know anything much about the distribution of the observations, but we can still get a sample mean (X-bar). We will never know what the actual sampling error is but we can make all kinds of probabilitic statements about it. For example,

1) X-bar gets closer to mu in a bunch of probabilistic ways as sample size increases. This means that as the sample size increases you can be certain that X-bar is getting close to mu so the sampling error goes to 0 (this is the law of large numbers).

2) With a few assumptions, the sampling distribution of X-bar - mu is normal with mean 0 and variance = sigma^2/n. This is remarkable and is the central limit theorem. Remember I don't know anything about the underlying distribution of the observations. Thus, I know the distribution of the sampling error so I can make any probabilistic statement about it.