Question

Respond to the following in a minimum of 175 words, please type response:

The standard error of the estimate of the mean is represented by the equation: σ√n Discuss what this equation means, using your own words and explain why we use it. Consider how it relates to the fact that we are making assumptions about the population and not just the sample.

Answer 1

Suppose we are working with some distribution, of which we do not know the population mean . So we use the statistic (sample mean) as an estimator for . Since this is only an estimate and not the true value, we should also deal with the errors associated with it.

This error is calculated as . So basically by using the term standard error, we express the possible variations that can occur while estimating from the statistic .

Now suppose we have i.i.d. observations drawn from some distribution with population mean (which is unknown and we want to estimate)and population standard deviation . So we have,

Now to estimate the variation in our statistic , we calculate

So we have, . This is how we get that standard error of estimate of the mean is . It indicates that the observed value of will vary from sample to sample, and its standard deviation is .

• How does it relate to the fact that we are making assumptions about the population and not just the sample?

In practical purposes, we shall work with a single sample and the true value of will be unknown to us. Given a sample, we can calculate the sample mean, and we shall use that as an estimate for population mean. But notice that, if we draw a different sample, the sample mean may come out to be totally different. In other words, although the parameter is a constant number, the estimator is a random quantity. Given one sample, the observed value of has some error associated with it as an estimate for . But in calculation of the standard error, we did not use anything about the sample, we only used distributional assumptions. So the standard error term is related to assumptions about the population and not just the sample.