Statistics and Probability-Part 1

Random Sample A sample is a group of people or set of objects about which you want to gather information. You can collect data by questioning a sample of people from a specific population or by examining a sample of objects from a set that has like characteristics. When taking a sample, it is important to do so randomly so that every member of the population has an equal chance of being selected. You can also collect data by designing and running an experiment or by carrying out a simulation.

When collecting data from a sample, you should avoid bias When collecting data from a sample, you should avoid bias. Some possible causes of bias include: -Incorrectly choosing the sample. -Neglecting to account for the people who do not respond -letting interviewers select the people that they want to interview.

A summer band camp has middle school students from all 50 states and Washington, D.C. Twenty students from Delaware are at the camp. The census bureau data indicates that about 80% of the population in Delaware lives in urban areas. Do you think it is likely that of the 20 middle school students from Delaware at the band camp, 16 live in urban areas? Explain your reasoning. Sue states, “Only eight of the 20 Delaware students in the band camp live in urban areas.” Does this number surprise you? What are some possible reasons for the rather low number?

The question to investigate is, “How likely is it that in a randomly selected sample of 20 middle school students from Delaware, only 8 of them live in urban areas?” Choosing a random sample is important because it helps reduce bias in the sampling process. A sample is random when it favors certain outcomes or some parts of the population over others. Care must be taken so that any member of the population has an equally likely chance of being chosen in the sample. In statistics, fair sample means that each element of a set has an equal probability of occurring.

What is meant by a “randomly selected sample” of students from Delaware? How could someone randomly select 20 middle school students from Delaware? Suppose you had a random sample of students from Delaware. How many of them do you think would be likely to come from urban areas? To investigate this question, you can create a model, or a simulation of the situation in Delaware.

You are given rectangles that represent the percentage of people form urban and rural areas in Delaware, where two out of every ten people are from a rural area. Take out a rectangle. Record in a table what is written on the rectangle and put the rectangle back in the bag or box. Shake the container to mix the rectangles. Repeat 20 times. Record the data that is drawn in the table below

How many of the students in the class have exactly eight students from Delaware who live in an urban area in the sample? Is this result surprising? Why or why not? How many of the students in the class have exactly 16 students from Delaware who live in an urban area in their sample? Is this result surprising? Why or why not? It is often easier to get a clear picture of the data if you have a graph. Create a histogram using the data from above.

Based on the histogram, what do you notice about the number of Delaware students in a sample of 20 students who live in the cities? Based on your data, how likely do you think it is to have 14 to 18 Delaware students in a random sample of 20 students who live in urban areas? If this simulation were repeated, do you think the data would be the same?

Biased Sampling Many samples are biased because they favor certain outcomes or they favor some part of the population over others. In such instances, there is a systematic error in the way the sample represents the population. Consider the following situation: Some TV stations poll the public. Viewers are urged to call specific numbers to voice their opinions. Dialing one number registers a “yes” vote; dialing another number registers a “no” vote. What are two problems with this type of sampling?

Bias can result when underlying factors about a situation are not considered during the selection of a sample. Why might the information collected about students at a band camp be biased?

Read the following survey situations Read the following survey situations. Explain how the poll could involve bias. The chief of police in a major U.S. city wants to determine how the public feels about the department. He prepares a questionnaire and sends police officers out to interview people in randomly selected sections of the city.

A magazine for health foods and organic healing wants to establish that large doses of vitamins improve health. The editor asks readers who have regularly taken vitamins in large doses to write the magazine and describe their experiences. Of the 2,754 readers who reply, 93% report some benefit from taking large doses of vitamins. A researcher wants to find out how many Americans intend to vacation in the United States in one year. To avoid bias, she selects 27 travel agencies in large cities and interviews each seventh visitor. The results of her research are published and titled, “Record Number of Americans to Foreign Destination”