David B. Howell
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A. Objectives—students will be able to
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1. define sample;
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2. describe and give examples of random sample;
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3. Iist factors compromising randomness;
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4. describe variations among samples;
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5. describe how changing sample size affects variation among several samples;
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6. select a sample size to estimate a population characteristic with confidence;
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7. discuss ways to quantify confidence.
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B. The experimental question—What percent of the colored cubes in the box are red? (or colored paper squares)
C. Issues, and some possible resolutions—
[Materials
: A box of at least 1500 objects; some of them are red. I used a box of commercial “cube-o-grams,” which are cubic centimeters of various colors. Twenty sheets of various colored construction paper cut into square inches would do just as well. So would a lot of M&Ms, though the sampling procedures would have to be sanitized. It is important that the objects be essentially identical except for color and easy to scramble so that a sampler, reaching into the box without looking, can select a random sample. It is interesting, but not essential, to have a variety of colors represented. The teacher will be more convincing if (s)he doesn’t know the exact percentage of red ones, either.
At least 3 copies of Worksheet 1 per student.
At least 2 sheets of graph paper per student.
With this “pure” example we will try to make explicit the ideas developed in earlier lessons. We can examine sampling and results without subjective distractions (“I don’t care what happened; I like pepsi best!”) Hopefully, we have done enough unverifiable predicting from samples earlier that we can resist the temptation to verify here. Use of a bigger population reduces the temptation, of course.]
Here in this box are several thousand residents of the planet Colsquar. Weird little creatures . . . , they look like this [display one]. They come in many colors; it is easy to assign them to different groups based on their color just as we can assign people in America to different groups based on income, or years of school, or nationality of ancestors, or race or religion, etc. Anyway, let’s pretend that we work for a company that produces rock music hits on bright red records. Red residents of Colsquar like red records! The question is—What is our potential market? What percent of the population of Colsquar is red?
Each of you should take a sample of 10 objects. [It makes an interesting side trip to have students predict just what classes, or colors are in the total population. With my set of cubes there are at least 12 colors. How many samples of 10 need be pooled to get at least one cube of each color? Generally, about 5, or a sample of 50, is sufficient; it appears that a couple of colors each represent only 2-4% of the total.] Record your results on Worksheet 1. Now take a second sample of 10. Record it. And a third sample.
[Here is an example of a student’s worksheet at this point:]
(figure available in print form)
Let’s make a list of several samples of 10 from the class. List on the board or overhead. Here is an example:
(figure available in print form)
What would you predict is the percent of red in the total population? (10%. 5 to 40%. 20%. Less than 50%. More than 10%. How about the average of all of those?)
Let’s make a graph—a histogram, actually—of all of our samples of 10.
[Here is an example:]
(figure available in print form)
Does the histogram help you predict?
On your worksheet, add the results of each sample of 10 so you have a record for a sample of 30. [Referring to the DH samples earlier, we’d have
(figure available in print form)
Let’s graph several samples of 30.
[Here is an example:]
(figure available in print form)
What do you want to predict now? Does anyone want to predict less than 7%? More than 17%? How
sure
are you of your prediction range?
Let’s pool samples of 10 into samples of 50. List 20 or 30 groups of 50, drawn from various 10-samples from the class, on the board. How graph this.
[Results might look like this example:]
(figure available in print form)
What prediction now? How sure are you? Are there any results you want to rule out as not likely?
List the range [low to high] for your samples of 10. And the mean. How do the same for the class list of samples of 30. And the class list of 50s.
[Here is an example of the results:]
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Sample size
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Range (percent of red)
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Mean
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10
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0Ð40%
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13%
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30
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0Ð23%
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13%
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50
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6Ð22%
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13%
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D. Observations and discussion to Objectives—
What happens to the range of possible results of several samples as the sample size increases? (It decreases.) Would you have expected that? Why or why not?
What happens to the mean of several samples as the sample size increases? (It doesn’t change much.) Would you have expected that? Why or why not? [The Central Limit Theorem is being illustrated here.]
On the basis of
one
sample of 50, what would you be willing to predict about the percent of red cubes in the total population? How sure are you?
[In the answers to the last set of questions, we have done at an intuitive level exactly what the political pollsters and other professional samplers do. We have taken a small random sample of the population (50 of a few thousand) and predicted that the total population has a characteristic (red) at, say, the 13% level, plus or minus 5% (range of 8 to 18%) or perhaps even smaller. We have made that prediction with considerable confidence (unquantifiable just yet) based on our examples.]
The series of Lessons is concluded. Students should have met the Objectives. There are plenty of follow-up activities possible. One might wish to do more, similar, sampling activities around research questions students pose. What percent of a population is left-handed? Owns a Michael Jackson record or tape? Plans to attend college? It might be valuable to pursue other topics suggested in discussions of the activities above. One might want to return to probability, for example, to begin to build a more formal set of tools for analysis of these same activities. Wherever . . . The base of experience established here should make the task easier for both student and teacher.