When we estimate quantities of interest, we also estimate their standard errors.

Question

When we estimate quantities of interest, we also estimate their standard errors. For instance, when we estimate the average height of college students, we use a random sample of students: we compute the sample mean height, sample standard deviation, and use that to compute the standard error of the sample mean. Similarly, for the unknown slope parameters , we compute the estimated values ˆ and their standard errors. The process of computing a standard error (and the related confidence intervals) relies on the logic of repeated sampling. a. What is the logic of repeated sampling? b. How do you use thus interpret the standard error? The 95 percent confidence interval?

Explanation / Answer

Answer

a) Repetition is done to detect outliers. It is possible that in an experiment a particular value in a particular treatment goes awry which can happen. If repetition is done, this possibility can be immediately detected and the erroneous value can be eliminated so that it does not lead to wrong inferences.

b) All estimates are averages. That is why we say sample mean is an unbiased (expected value) estimate of the population mean. But, the actual population mean may not be equal to the sample mean. In fact, more often than not, it wont be. So, to obviate this, a range is given alongwith the estimate using SE. As a rule of thumb, the actual value of the parameter would lie between estimate plus/minus one SE in about 67% of times, between estimate plus/minus two SE in about 95.6% of times, and between estimate plus/minus three SE in about 99.7% of times, in fact, Statistics strictly prohibits giving only estimates – estimate must always be given in conjunction with its SE so that reliable decisions can be taken.

c) Confidence Interval is a more refined way of expressing the above rule of thumb. It enables to arrive at a reliable range for the estimated parameter alongwith a confidence level which indicates ‘how sure’

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