Do I have to take many samples from my population to create a sampling distribution of the mean in order to infer what the population mean is likely to be? If I need only one sample, how does that work (to infer the population mean)? Thanks.
Do I have to take many samples from my population to create a sampling distribution of the
Basics of sampling distribution of mean:
The sampling distribution of the sample mean is essentially a probability distribution. Evidently, the purpose of sampling is that you do not already have a lot of information regarding the population and wish to gather knowledge of the population by means of sampling.
Now, the probability distribution of the mean is not known to you. You want to create the probability distribution. For this, you need to first set the desired sample size, say n.
Then, you have to collect all possible samples of size n from your population.
For each sample, calculate the sample mean.
Now, for each unique value of the sample mean that you found from your many samples, find how likely it is. If you get, say, s possible samples of size n from your population, then you will get s sample means, say m1, m2, …, ms. All these s values will not necessarily be unique. A few of the values may come once; others may come twice, thrice, etc. For every value occurring once among the s values of means, the probability will be 1/s. Similarly, for every value occurring twice or thrice, the respective probabilities will be 1/2s or 1/3s and so on.
If you organize these unique values of means with their respective probabilities, you will get the probability distribution of the sample mean.
If you take the expectation of this probability distribution of the sample means, you will get the population mean.
So, yes, you have to take many samples from your population to create a sampling distribution of the mean in order to infer what the population mean is likely to be.
Sampling distribution of the sample mean:
Now, it can be easily shown that the expected value of the sample mean is the population mean, with a standard deviation of the standard deviation divided by the square root of the sample size. In case you have no idea of the population standard deviation, you can calculate the standard deviation, say, s from the sample.
Thus, in theory, if you need only one sample of size n, then you collect only one sample and find its mean (m) and standard deviation (s). Then, you can say that the population mean is likely to be the same as this sample mean you found, although there is likely to be a deviation of (s/√n) about this value.
In other words, you can say using your one sample (of size n), that it is most likely for the population mean to lie within an interval of length (s/√n) both above and below the sample mean, m, that you found.
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