Population and Samples

In this lesson, you will extend your knowledge of statistics from last year as you learn to use random samples to make predictions about an entire population and judge the possible discrepancies of those predictions. You will use real-life situations to show the purpose for using random sampling to make predictions (inferences) about a population.

The main purpose for using sampling is to gather data in order to learn something about a population. The most important type of sample is a representative sample, which is a part of the population that is similar to the entire population. Random samples are those that best represent the population and in these samples, every item, person, or object in the population has an equal chance of being selected. A biased sample does not fairly represent the population because this sample allows some members of the population to have a greater chance of being selected than other members. In order to get a better estimate, get enough samples. When samples are too small, they do not represent the population accurately and the inferences made from them will be incorrect, so make sure the random sample is as large as time and resources allow.

Surveys are often used to gather data and are made up of a question or set of questions. Be careful because surveys can be biased in the way the questions are worded. When creating a survey, always try to be precise, brief, and offer no opinions in the wording.

First, examine some methods for selecting representative random samples when you gather data:

Self-Selected – Ask the entire population for volunteers (i.e. respond to an email or fill out a survey form from a public place).

Systematic – Use a pattern in the selection. For example, survey every 10th person entering the school.

Stratified – Divide the population into subgroups…boys and girls…seventh and eighth graders; and randomly select equal participants from each of those groups.

Random Method – Draw names out of a hat, where all have a chance to be selected.

Now, here are a few ways of selecting a sample that might be biased:

Select a specific group of people, such as football players or honor roll students.

Convenience – Choose people who are easy to contact, such as close friends, classmates, or those sitting near you in the cafeteria.

Allowing people to volunteer can also be biased if you do not open the survey to the entire population and make access the same for all of the population.

Example

Adam feels that more students in his school like the strawberry smoothies than those who favor the mixed berry smoothies. He surveys the students in his class and discovers that 22 of the 26 students in his class do prefer strawberry. Do the results of Adam’s survey support his inference that more students at the school prefer the strawberry smoothie to the mixed berry one?

First, a sample should be representative of the population and the population is the students in Adam’s school.

Though the students in the class are part of the population being studied, and probably represent it well, this sample is a convenience sample because every member of the population did not have an equal chance of being selected. Adam met part of the definition of a representative sample, but not enough to be an accurate random sample.

The results of this survey might support his prediction, but random samples are preferred when gathering information about the population.

Example

Phyllis asked every seventh student entering the middle school if he or she preferred a dog or a cat as a pet. Can the results of her survey be used to make inferences about students’ favorite pet at the school?

In a random sample, remember that everyone in the population has an equal chance of being part of the sample and in this case, every student entering the school did have the same chance of being one of the seven.

Is the sample representative of the population? Since the students in the school are the population and the random sample is from the student, then it is definitely representative of the school population.

Is the survey biased? In a way, this type of survey can be biased because only dogs or cats were included. Other animals might be the favorite pet of the student population. On the other hand, these survey results can only be used to make predictions about student preferences for these two animals included in the survey.

Before we move forward, take a look at this video to see what constitutes a reasonable sample.

Now that you have learned about different types of random samples, it is time to use random samples to make mathematical predictions, using proportions.

Example

Kristin chooses a random sample of 50 out of 400 students. She found that 8 of the students had been to a Justin Bieber concert. She claims that, based on the data from her sample, over 50 of the 400 students have been to a Justin Bieber concert. Let us investigate.

We can write and solve a proportion.

Based on her survey, it is likely that she is correct.

Many companies use random sampling to purchase supplies. For example one department store receives a shipment of 2,000 glasses. Out of a random sample of 20 glasses, 4 are broken. How many glasses can they expect to be broken in the entire shipment?

Yikes, they can expect to have about 400 broken glasses in the shipment!

Look at these examples of how you can use proportional reasoning to make inferences.

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