Estimation

If a population mean μ is to be estimated by (1) using simple random sampling, with replacement, (2) drawing a sample of size n (where n is of at least moderate size), and (3) using the sample mean x as the estimate, then X (the result of the estimation procedure, viewed as a random variable) is approximately normally-distributed with E[ X ] = μ and StdDev[ X ] = σ/√n , where σ is the population standard deviation. Consequently, the estimation procedure has roughly a 95% chance of yielding an interval

which contains the population mean, where s is the sample standard deviation (an estimate of σ).

Estimation Terminology

the standard error of the mean (i.e., one standard-deviation's-worth of exposure to sampling error, when estimating the population mean)
the margin of error of the estimate (at the 95%-confidence level) when the sample mean is used as an estimate of the population mean
a 95%-confidence interval

Estimating a Proportion

In order to estimate the fraction (percentage, proportion) of individuals in the population who possess a certain characteristic, let those individuals with the characteristic be labeled as ‘1’s, and those without the characteristic as ‘0’s, and estimate the population mean of this artificial (“dummy”) variable. (The mean is the number of ‘1’s, divided by the population size – This is the proportion sought.)

the sample proportion (mean)
the sample standard deviation
the standard error of the proportion
the margin of error of the estimate

Proper Perspective

Estimates are made in order to be used in further decision analysis. (E.g., should the furniture manufacturer buy advertising space in this magazine?) The margin of error in the estimate is used for a preliminary decision: Can the estimate be trusted enough to use it in the further analysis? (The margin of error also plays a role in subsequent sensitivity analysis.)

Therefore, it is sufficient to know roughly how large the margin of error is, and the material in the next section is more of theoretical interest than of practical importance.

Two Technical Details

(1) Simple random sampling is usually done without replacement (so there is no chance of the same individual appearing twice in the sample). In this case, there is a bit less exposure to sampling error than in sampling with replacement.

the standard error of the mean (when sampling without replacement from a population of size N)

In most managerial applications, the population size is much larger than the sample size; frequently the population size isn't even known. As long as N is much larger than n, the final factor in the equation above is very close to 1. And in all cases, it's no more than 1. Therefore, omitting it from our calculations, and treating sampling with and without replacement as being the same, is completely benign, and is usually done in practice.

(2) If the sample size is much below 20, the assumption that the sample mean is approximately normally distributed is questionable. In general, a 95%-confidence interval for the estimate cannot be given: The estimate should just be reported along with a warning that the sample size was quite small. (The actual sample data can be attached to the warning, since it will fill less than a single page.)

However, if the population itself is approximately normally distributed, then a 95%-confidence interval for the estimate can be reported. Still, in both this special small-sample case and in the general larger-sample case, we're cheating a bit when we use s instead of σ. Taking this into account, we need to make a slight adjustment in the margin-of-error formula:

the margin of error of the estimate (at the 95%-confidence level), where x.xxx comes from the t-distribution with n-1 degrees of freedom (see the table below)

Note that, as the sample size grows, the correct “approximately 2” multiplier becomes closer and closer to 1.96.