In this activity, students begin by practicing their understanding of proportions and probabilities by examining the data set they have available. In the fourth problem, students obtain a sample from the population using tools they choose (MP5) and examine the sample they selected to compare it to the expected proportions and probabilities calculated in the first three problems.

The problems are intended for students to use their own data set to answer. Although they are kept in pairs for the entire lesson, this activity should be done individually.

The purpose of this discussion is to connect the ideas of probability and random sampling from the unit.

Consider these questions for discussion:

• “How is selecting a sample at random connected to probability?” (A random sample should give each value an equal chance of being chosen. Therefore, each value has a probability of being chosen.)

• “How could we simulate the probability of getting at least 2 values in the sample of 10 from Springfield Middle School?” (Since 20% of the values come from Springfield Middle School, we could put 10 blocks in a bag with 2 colored red to represent Springfield Middle School. Draw a block from the bag, and if it is red, it represents a score from Springfield Middle School. Replace the block and repeat. Get a sample of 10 and see if the sample has at least 2 red blocks. Repeat this process many times, and use the fraction of times there are at least 2 red blocks as an estimate for the probability that a random sample will have at least 2 scores from Springfield Middle School.)

In this activity, students practice estimating a measure of center for the population using the data from a sample. The variability is also calculated to be used in the following activity to determine if there is a meaningful difference between the measure of center for the population they used to select their sample and the measure of center for another population.

The purpose of the discussion is for students to make clear their reasoning for choosing a particular measure of center and reiterate the importance of variability when comparing groups from samples.

Consider these questions for discussion:

• “Which measure of center did your group choose? Explain your reasoning.” (A median should be used if there are a few values far from the center that overly influence the mean in that direction. If the data is not approximately symmetric, a median should be used as well. In other cases, the mean is probably a better choice for the measure of center.)
• “Why is it important to measure variability in the data when estimating a measure of center for the population using the data from a sample?” (To use the general rule, the difference in means must be greater than 2 MADs to determine a meaningful difference. If there is small variation, then the samples may have come from a population that also has a small variation, so differences among groups may be more clearly defined.)

In this activity, students use the values computed in the previous activity to determine if there is a meaningful difference between two populations. Following the comparison of the groups, students are told that the populations from which they selected a sample are identical but shuffled. Students should use their understanding of sampling to construct an argument for why their means are not exactly the same, but reasonably close (MP3).

Ask each group to share whether they found a meaningful difference.

Tell students, “With your partner, compare the starred data for the two groups. What do you notice?”

Tell students that the two populations are actually identical but rearranged. Ask, “Did any groups get different means for your samples? Explain why that might have happened even though the populations are the same.” (Two random samples from the population will usually not contain the same values, so different means are probably expected.)

One thing to note: The general rule is designed to say whether the two populations have a meaningful difference or if there is not enough evidence to determine if there is a meaningful difference. On its own, the general rule cannot determine if two populations are identical from only a sample. If the means are less than 2 MADs apart, there is still a chance that there is a difference in the populations, but there is not enough evidence in the samples to be convinced that there is a difference.