In Section 7.1, we gained some intuition about sampling and its characteristics. In this section, we introduce some statistical definitions and terminology related to sampling. We conclude by introducing key characteristics that will be formally studied in the rest of the chapter.
Subsection7.2.1Population, sample, and the sampling distribution
A population or study population is a collection of all individuals or observations of interest. In the bowl activities the population is the collection of all the balls in the bowl. A sample is a subset of the population. Sampling is the act of collecting samples from the population. Simple random sampling is sampling where each member of the population has the same chance of being selected, for example, by using a shovel to select balls from a bowl. A random sample is a sample found using simple random sampling. In the bowl activities, physical and virtual, we use simple random sampling to get random samples from the bowl.
A population parameter (or simply a parameter) is a numerical summary (a number) that represents some characteristic of the population. A sample statistic (or simply a statistic) is a numerical summary computed from a sample. In the bowl activities the parameter of interest was the population proportion \(p = 0.375\text{.}\) Similarly, previously a sample of 50 balls was taken and 17 were red. A statistic is the sample proportion which in this example was equal to \(\widehat{p} = 0.34\text{.}\) Observe how we use \(p\) to represent the population proportion (parameter) and \(\widehat{p}\) for the sample proportion (statistic).
The distribution of a list of numbers is the set of the possible values in the list and how often they occur. The sampling distribution of the sample proportion is the distribution of sample proportions from each possible random samples of a given size. To illustrate this concept recall that in Subsection 7.1.3 we drew three histograms shown in Figure 7.1.12. The histogram on the left, for example, was constructed from taking 1000 random samples of size \(n = 25\text{,}\) then finding the sample proportion for each sample and using these proportions to draw the histogram. This histogram is a good visual approximation of the sampling distribution of the sample proportion.
Be careful as people learning this terminology sometimes confuse the term sampling distribution with a sample’s distribution. The latter can be understood as the distribution of the values in a given sample.
A histogram from a simulation of sample proportions is only a visual approximation of the sampling distribution. It is not the exact distribution. Still, when the simulations produce a large number of sample proportions, the resulting histogram provides a good approximation of the sampling distribution. This was the case in Subsection 7.1.3 and the three histograms shown in Figure 7.1.12.
The lessons we learned by performing the activities in Section 7.1 contribute to gaining insights about key characteristics of the sampling distribution of the sample proportion, namely:
The first two points relate to measures of central tendency and dispersion, respectively. The last one provides a connection to one of the most important theorems in statistics: the Central Limit Theorem. In the next section, we formally study these characteristics.
The population parameter is the true proportion of red balls, \(p\text{.}\) We can know it exactly by counting all balls and computing the proportion in the full population. In this case, \(p = 900/2400 = 0.375\text{.}\)
