Different samples from the same universe will give different estimates of the universe value. he estimate from a particular sample will generally differ from the universe value because of sampling error – that is, because the sample chosen will not be a precise replica of the universe. If researchers took another random sample from the same universe, the resulting estimate could differ a little, somewhat, or a great deal from the estimate obtained from the first sample. They are then faced with determining how precise their sample estimates are.
After a single sample has been taken, investigators typically would like to determine a range of values within which they could be quite sure that the true universe value lies. They would like, also to be able to specify quantitatively how confident they are that the indicated range of values will in fact, include the true universe mean.
What is wanted here is called a confidence interval estimate. After a sample has been taken, the researchers wish to say with a particular degree of confidence, that the true universe mean lies between two specified numerical values, called confidence limits.
A Commonly encountered Confidence Interval Estimate:
According to a poll conducted by Wall Street journal /NBC News, 56% of Americans feel President Reagan is too set in his ways to change his management style.
Chances are 19 out of 20 that if all households with telephone in the US have been surveyed using the same questionnaire, the results would differ from the poll findings by no more than three percentage points in either direction.
The implied confidence interval here is 56% ±3% with confidence limits of 53 percent and 59 percent. The writer is ’95 present confident (chances of 19 out of 20) that between 53 percent and 59 percent of all Americans feel Reagan is too set in his ways.
The theory of probability sampling provides methods for establishing such confidence interval estimates. This permits the researchers to evaluate the ‘precision’ of a sample based estimate. Stated another way, with simple random sampling researchers can calculate confidence limits within which the universe value being estimated will probably lie.
Several concepts are needed to develop the idea of confidence intervals. The following discussions present these concepts, the first of which is the distribution of sample means.
The distribution of Sample Means:
It has been emphasized that different samples from the same universe will lead to different estimates of the universe mean or universe percentage. For examples, assume one chooses all possible simple random of two from a universe of six persons in s department store. Each person is carrying the following amount of cash: A – $20, B-$80, C- $100, D- $100, E-$100, and F-$200. The universe mean is $100. Seven different ample means ranging from $50 (sample AB) to $150(samples CF, DF, and EF) would be obtained with the following relative frequencies.
Such a listing of all possible means, together with their relative frequencies if occurrence is called a sampling distribution of the mean or for short distribution of sample means. In this case, it is the distribution of sample means, for simple random samples of size two, from the department store universe of six persons.
Predicting the variability of samples means: Given the distribution of sample means (as in the above illustration) one can predict how thee estimates will vary relationship to the universe mean being estimated. In the illustration if repeated samples of size two are chosen, then 44 percent (7 out of 15) of sample means will lie between $90 and $110 or within $10 of the universe mean of $100. Similarly it would be expected that 73% (11 out of 15) of samples would be located between $60 and $140 or within $40 of the universe mean.
Knowledge of the sampling distribution of the mean enables the researcher to measure the precision of the estimate obtained from a single sample. If one knew only that a sample estimate would vary – but had no information as to how it would vary it would be impossible to assess the sampling error of that estimate.