6.1 Samples and Populations
- Key Idea 1: A sample is believed to be representative of something larger, namely, the population it has come from...
Discussion
- When we take a sample, such as a sample packet of a new brand of detergent, we are not usually interested in the sample for its own sake. Rather, we are interested whether a full-sized box of the new detergent will repeatedly produce clean laundry.
- Here are some examples
- A national poll of 1500 people is used to determine whether people are positive, neutral or negative about the economy in the next year.
Sample: the 1500 responses
Population: All Americans
- One hundred family units from Los Angeles are sampled and their family incomes recorded.
Sample: the 100 family units
Population: all family units in Los Angeles
- A grain-elevator operator takes five randomly selected containersful of corn from a truck being unloaded and tests it for moisture content.
Sample: the five containersful of corn
Population: the truckload of corn
- We count number of cures achieved by a drug when tested on 10 patients.
Sample: 10 patients
Population: all the people (possibly very many) who have or will ever contract the disease in question (this population is unbounded in size)
- Key Idea 2: When we have a randomly-selected sample from a larger population, we can use our measures of sample data center and spread as estimates of the center and spread of the population...
Discussion
- These quantities (center and spread) are termed population parameters (the word parameter is NOT used with respect to the same measurements in a sample).
- Using the sample mean, we estimate (with error) the population mean.
- The variance of the sample will help us in two important ways:
- First, it estimates a parameter called the population variance.
- Second, it helps us assess how accurate the sample mean is in its role of estimating the true population mean.
- Note: Until now we have used the term mean to refer to the average of a sample.To keep things straight, henceforth we will use the terms sample mean and population (or theoretical) mean.
Example