Why are sampling distributions important to the study of inf

Why are sampling distributions important to the study of inferential statistics? In your answer, demonstrate your understanding by providing an example of a sampling distribution from an area such as business, sports, medicine, social science, or another area with which you are familiar. Remember to cite your resources and use your own words in your explanation.

Solution

A statistic has its own distribution, associated but distinct from the \"usual\" population distribution one thinks of. When one calculates a statistic from some data, you may intuitively think of it as being fixed. But while that measurement is fixed, the statistic has variability and thus has a distribution, because it is based on a sample. In layman\'s language: Basic inferential statistics is all about figuring out what\'s real and what\'s noise; the distribution represents noise and needs to be accounted for.

Inferential statistics is defined as the branch of statistics that is used to make inferences about the characteristics of a populations based on sample data.

One of the most important concepts in inferential statistics is that of the sampling distribution. That\'s because the use of a sampling distributions is what allows us to make \"probability\" statements in inferential statistics.

Typically, there is an assumption that every data point collected can be modeled as an independent and identically distributed random variable from the population distribution, F.  
Often (especially in introductory stats), we assume that we know something about F (e.g. that it\'s normal or exponential or Poisson) but there are still things about F that we\'d like to infer (like the variance of the normal or the mean of the Poisson). These unknown details about the distribution are described as unknown parameters.

So what can we do with the data (i.e. the IID random samples from F) to help us infer things about F? Commonly (but not in all branches of stats), we create estimators. An estimator is really just some function of the samples (hence it, like the samples themselves, is a random variable) that we think should take on values pretty close to the unknown parameter we want to estimate (e.g. the mean of F or its variance). But if the estimator is a random variable, it must have a distribution. We need to figure out something about this distribution if we expect to learn very much about the unknown parameter.