In order to decide whether you like a particular soup, do you need to eat the whole pot? Usually, you can get a good sense of the flavor of the soup by taking a small taste, or a sample.
When conducting a study, it is usually not possible to survey every person or object in the population in which you are interested (for example, all the residents of the United States, all the students at your school, or all teenage shoppers). However, just as you can learn about the flavor of a soup with a small taste instead of eating the whole pot, you can learn about a population by sampling instead of taking a census. Unfortunately, large groups of people are much harder to stir than pots of soup to get a representative mix of their populations.
In this lesson, you will consider ways that statisticians do the equivalent of “stirring” populations in order to take samples that represent the whole population well.
As you work with your team today, consider the following questions.
How should we decide what is “typical” of a group?
How do random selection and intentional selection compare?
9-16. How should you choose people for your survey to make up a sample that you think represents the population for your research question?
a. Discuss this question with your team and write down your ideas. Be sure to justify why your ideas will help you choose a representative sample.
b. Do you think your sampling method is better than randomly selecting members of the population? Why or why not?
9-17. Some astronomers believe that a number of small objects, called Vulcanoid asteroids, exist between Mercury and the Sun. Other scientists, however, do not believe that these asteroids exist. The brightness of the sun has prevented direct observations, so the controversy remains unresolved.
Imagine the following scenario. Scientists invent a special camera that can photograph these asteroids, even though they are so close to the sun. They put the camera on a satellite and send it off. To their joy and amazement, the satellite begins to transmit images back to the earth and the asteroids are visible! Vulcanoids are real!
Unfortunately, in just five seconds, the heat of the sun destroys the camera and the transmission stops. Due to a technical malfunction, the five seconds of video images were not recorded. The scientists who were watching have to rely on their memories of the images.
. Help the scientists determine the size of a “typical” Vulcanoid asteroid by modeling this scenario. First, look at your “video” picture (Lesson 9.1.2A Resource Page) for exactly five seconds. Then,without talking to anyone, write down your estimate of a typical asteroid diameter in millimeters.
a. Follow your teacher’s instructions to contribute your estimate to the class’s histogram. Sketch the histogram on your paper and describe its attributes, including the center, shape, spread, and the presence of any outliers.
9-18. Now suppose that the camera on the satellite and the recording equipment were repairable and the scientists were able to record a good still photograph of the Vulcanoid asteroids. Now you have much more reliable information with which to determine the size of a typical asteroid.
. Look at your photograph on the Lesson 9.1.2A Resource Page. Again, without talking to anyone, pick ten “typical” asteroids and measure the diameter of each in millimeters. Write down each of your measurements and then calculate the mean diameter. Do you think this result is a better estimate of the size of a typical asteroid than the one you estimated in problem 9-17? Why or why not?
a. Again, follow your teacher’s instructions to contribute your calculated mean to the class’s new histogram. Sketch the histogram on your own paper and describe how this histogram compares to the histogram of the class’s estimates from problem 9-17.
b. In general, which set of estimates is more accurate? How can you tell? Be sure to use your comparison of the histograms to help justify your answer.
9-19. You will use your calculator’s ability to generate random numbers or a random number generator to help you to randomly select 10 asteroid images.
. First, your teacher will give you instructions to make sure that your calculator does not generate the same random numbers as someone else’s calculator.
a. Your teacher will now give you instructions to use your calculator to randomly select ten asteroids. It is possible that some will be repeated. Each member of your team should generate his or her own set of random numbers.
Measure the ten asteroids that were randomly chosen by your calculator. If you have repeated numbers, measure those asteroids twice. Calculate the mean diameter of your ten asteroids.
b. Again, follow your teacher’s instructions to contribute your calculated mean to the class’s new histogram of randomly selected asteroids. Sketch the histogram on your own paper and describe how this histogram compares to the histograms of the class’s estimates from problems 9-17 and 9-18.
9-20. HOW SHOULD SAMPLES BE CHOSEN?
Based on the data collected by your class, discuss the following questions with your team. Be prepared to contribute your ideas and questions to a class discussion. . Which selection method produced the most consistent (least variable) results? Why does this result make sense?
a. Look in particular at the differences in data produced by intentionallychosen samples and randomly-chosen samples. Why do the differences make sense? Be sure to consider differences in center, shape, spread, and outliers.
b. Which sampling method seems to produce the best estimate of the size of a “typical” asteroid? How can you tell?
c. Your teacher will provide you with the actual measures of central tendency of the sizes of the asteroid images. Does your idea about which sampling method is best change?
9-21. CHOOSING YOUR SAMPLE
Talk with your team about what it would mean to randomly choose a sample from the population in your survey (problems 9-1 and 9-6). How might you do that? Could your calculator help you? Will it be possible to survey a randomly-selected sample? Why or why not?
Store your survey questions in a safe place. They will be needed later in this chapter.