Your students have had an opportunity to investigate choosing samples. Now is the time for them to apply their knowledge to a particular experiment.
Materials: A paper bag and 84 connecting cubes (56 blue and 28 yellow, or use whatever colors are available), counters, or other small objects
Preparation: Put the 84 cubes into the paper bag before class begins.
Prerequisite Skills and Concepts: Students should be able to set up a proportion and solve it.
- Say: Yesterday we introduced the idea of picking a representative sample from a population because it may be impossible to collect data from every member of a population. What can you tell me about the best size of a sample?
Students should say that if the population is fairly large, randomly selecting one tenth of the population is usually satisfactory. Point out to them that if the population is small, you may need more than one tenth of the total.
- Say: Choosing a sample which is representative of the population is an important step in making good predictions. Biased samples are ones which don't represent the population and often lead to incorrect predictions.
- Bring out the paper bag with the 84 cubes in it.
- Say: This bag contains 84 cubes. Some of them are blue and some are yellow. We want to try to predict how many there are of each color by selecting a sample. Who would like to come up and select a sample for us?
- Have a student come up and pick four cubes from the bag without looking. Place the cubes so the whole class can see them.
- Ask: What do you think about the sample that was selected?
Students will probably say that the sample may not be representative because it is too small.
- Say: Yes, that's right. How could we have picked a better sample so that the results might be more reliable?
Students should suggest picking more cubes.
- Say: I will pick a sample which consists of 20 cubes.
- Look inside the bag as you pick 20 cubes, mostly all the same color. Say things such as, "not this one, yes, this is a good one," indicating that the sample chosen is not done at random.
- Ask: What do you think about the sample that I selected?
Students will probably say that you should not look in the bag when selecting the sample because the sample will be biased.
- Say: Who would like to come up and select 20 cubes?
- Have a student come up and pick at least 20 cubes without looking. Assume they chose 6 yellow and 14 blue cubes.
- Say: In our sample, we have 6 yellow and 14 blue cubes. Based on our sample, how many of the 84 blocks do you think are blue and how many are yellow? (Students may make some guesses.) Is there some way we might set up a proportion from our sample to predict the number of each color of cubes?
Students should suggest setting up the proportion 6/20 = x/84. If they don't, tell them that is how they could figure out the number of yellow cubes in the bag.
- Say: How would we solve this proportion to find x?
Students will probably say to cross multiply. (x = 25.2 or about 25) Have two students count out how many of each color are actually in the bag. Discuss the results with the class and repeat the experiment with a different number of cubes.
Wrap-Up and Assessment Hints
"What if" questions are good ways to assess the depth of a student's understanding. Consider the following problem: "If the mean of a data set is 38 and the numbers 48, 46, and 49 are added to the data set, what happens to the mean?" Students should know that the mean will increase. How much it will increase will depend on the number of items in the data set. For example, if there are 200 numbers in the data set, it will not increase very much. However, if there are only 5 numbers in the data set, then it will increase by more. Another question which would test the depth of understanding about the mean is: "If the mean for a set of data is 38 and a score of 49 is added to the data set, what score would need to be added to the data set in order to keep the mean equal to 38?" Students should recognize that the score of 27 is what would need to be added, since it is 11 points below the mean and 49 is 11 points above the mean.
Measures of Central Tendency:
Mean, Median, and Mode