**Introduction to Sampling Distributions and the ***t*-Distribution Study Guide

*t*-Distribution Study Guide

*The distribution of a random sample from a population is called the sample distribution. Using sample values, we can obtain estimates of the population parameters, such as the mean, the standard deviation, or a proportion. If we take another sample, it is very likely that the estimates from the second sample will differ from those of the first. Fortunately, this variation among estimates from different samples is predictable. The key to understanding this variation lies in gaining an understanding of sampling distributions (yes, these are different from sample distributions). Intuitively, it seems that as the sample size increases, we should do better. Sampling distributions in this lesson and the Law of Large Numbers in the next give us insight into what is meant by better.*

**Sampling Distributions Defined**

We learned that a parameter is a summary measure of a population, and a statistic is a summary measure of a sample. A statistic is some function of sample values that does not involve any unknown quantities (such as parameters).An important statistical idea is that parameters are fixed, but generally unknown, and that statistics are known from the sample, but vary. It is critical to always keep in mind whether we are thinking about a parameter or a statistic.

Suppose we are interested in the mean arm span of females attending college in the United States. (The arm span is the distance from the tip of one middle finger to the tip of the other middle finger when the arms are fully extended to the sides and perpendicular to the body.) Ten different people independently estimate the mean arm span by taking a random sample of 20 U.S. college females and finding the sample mean of the arm spans. It is very likely that this will lead to ten different estimates of the population mean arm span. Because each person selected a different sample, the arm spans of the people within each sample, and thus the sample means, will tend to be different. We have ten observations from the sampling distribution of the sample mean, one from each sample. The *sampling distribution* of a statistic is the distribution of possible values of that statistic for repeated samples of the same size from a population. That is, the statistic, the sample mean in our example, is a random variable, and the sampling distribution is the distribution for that random variable. Fortunately, we know quite a bit about the sampling distributions of the statistics that we will be most interested in.

**Example**

For each of the following circumstances, explain whether the quantity in bold is a parameter or a statistic.

- For a sample of 20 married couples, there was a difference of
**2.5**inches in the mean heights of the husbands and wives. - The Census Bureau reported that, based on the 2000 Census, the median age of residents of Oklahoma was
**35.3**years.

**Solution**

- The 2.5 inches is a statistic because it is based on a sample from the population of all married couples.
- The 35.3 years is a parameter. A census occurs when every unit in the population is contacted. The U.S. Census occurs every ten years and attempts to contact everyone in the United States.

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