Session 10, Part C:
Inferences and Predictions (30 minutes)

The NCTM (2000) data analysis and probability standards state that students should "develop and evaluate inferences and predictions that are based on data." In grades 3-5 classrooms, students are expected to develop and evaluate inferences and predictions, to propose and justify conclusions and predictions that are based on data, and to design studies to further investigate their conclusions or predictions.

Inference and prediction are more advanced aspects of working with data, as they require some notion of the ideas of sampling and population. Students in grades 3-5 are only beginning to develop an understanding of sampling. They often trust their own intuition more than the information they are obtaining from the data. Children begin to develop an understanding of these statistical ideas through conversations as they consider what the data are telling us, what might account for these results, and whether this would be true in other similar situations. Children's early experiences are often with census data -- e.g., the population of their class. When they begin to wonder what might be true for other classes in their own and other schools, they begin to develop the skills of inference and prediction. It is not until the later middle grades and high school that students begin to learn ways of quantifying how certain one can be about statistical results. Note 6

When viewing the video segment, keep the following questions in mind:

 • How does Ms. L'Esperance encourage students to make inferences and predictions? • What are some of the students' preliminary conclusions? • How are the ideas of sampling and population embedded in this conversation?

 Video Segment In this video segment, Suzanne L'Esperance facilitates a whole-class discussion as the students consider potential conclusions to the original problem on how large to build the house. Students discuss the variance in their data, the limitations of their small sample, and the need for additional data. If you're using a VCR, you can find this segment on the session video approximately 38 minutes and 33 seconds after the Annenberg Media logo.

Problem C1

Answer the questions you reflected on above as you watched the video:

 a. How does Ms. L'Esperance encourage students to make inferences and predictions? b. What are some of the students' preliminary conclusions? c. How are the ideas of sampling and population embedded in this conversation?

 Problem C2 Based on the family-size data gathered by the class and shown in the line plot below, how would you respond to Ms. L'Esperance's initial question: What size should she tell her friend to build his homes? What reasons can you offer to support this response, and how are they related to the ideas you have studied in this course? Are your reasons based on the data collected, or did you also bring in some of your own judgements?

Problem C3

Children are expected to develop and evaluate inferences and predictions. Evaluate each of the responses below by commenting on the following:

 • why the response makes sense (or doesn't) based on the data; and • the limitations of each response. In other words, what statistical ideas are the children not taking into account?

The children's responses to the question of how big to build the homes were as follows:

 a. "He should build homes for four people." b. "You can tell him to put in a couple of each, because some people live with two people, so he should put more fours and threes, but put some of the other kinds also." c. "He should build them for three people and four people." d. "I know some people that have six and eight people in their families, so he should build some larger houses too."

 Problem C4 In thinking about the data that were collected, in what ways might the students' sample be biased? How might you facilitate a discussion with the students about bias in data? What questions would you pose? What issues would you raise?

 Problem C5 According to the 2000 census, the average size of households in the United States is 2.62 people. How might your students respond to this information in light of their own data? What statistical ideas would you want to surface in this discussion?

Problem C6

If you were teaching this lesson on investigating family size, what questions could you ask students to encourage them to focus on each of these central elements of statistical analysis?

 • Defining the population • Defining an appropriate sample • Collecting data from that sample • Describing the sample • Making reasonable inferences relating the sample and the population