Lesson 5: Statistical Predictions Using One Variable

Lesson 5: Statistical Predictions in One Variable

Objective:

Students will devise a rule to determine how to choose a winner when predicting the typical height of all students in a large high school and measure the success of their prediction. They will consider different measures of success.

Materials:

  1. Heights of Students at a Large High School handout (LMR_4.4_HS Student Heights)

Vocabulary:

rule

Essential Concepts:

Essential Concepts:

Anyone can make a prediction. But statisticians measure the success of their predictions. This lesson encourages the classroom to consider different measures of success.

Lesson:

  1. Inform the class that for this lesson, our class will help judge a contest held at a particular high school. This school held a contest in which they selected students at random from a classroom and reported their height.

  2. The information in Steps 3 – 7 is included in the Heights of Students at a Large High School handout (LMR_4.4).

  3. They gave all of the students the data for the first 20 selected students, but you will not see these data! Each student was asked to predict the heights of the last 10 students. Here is the catch: students were allowed to give only ONE number that had to be used to predict all 10 heights.

  4. Three student teams made predictions about the height of the last 10 students. The judges of this contest want you to tell them how they should determine the winner.

  5. Your team’s job is to determine the winning team, the team that came in second place, and the team that came in third place. Your team must come up with two things:

    1. You must support your choice of a winner by using a rule for calculating a total score for each team. The rule must be applied to each team’s guess to determine their placement and your team must be able to explain how your rule helped select the winner.

    2. You must write instructions to the judges that explain how to use your rule to select a winner. For example, do they choose the team with the largest score? The smallest?

  6. Here are the predictions of the three teams:

    Team A: 69 inches

    Team B: 70 inches

    Team C: 66 inches

  7. Display Plot A, found on page 2 of the Heights of Students at a High School handout (LMR_4.4). This dotplot displays the heights shown in the Actual Outcome column of the table. Inform students that this dotplot and table is provided to help them come up with a method to determine a winner. The table is to visualize the predicted heights side-by-side with the randomly selected heights.

    Notes to teacher:

    1. Students may have to be reminded that negative values with large absolute value are larger than positive values with small absolute values (e.g., 10 is larger than 3).

    2. Let students struggle for a little bit. A prompt to get them started: Look at the difference between a team's prediction and the actual outcomes (e.g., for the first height, Team A predicted 69, actual outcome was 63, so 69-63=6). They might also need to be nudged towards the sum of these differences – they need to produce a single score, not 10 separate scores.

    3. Here are some rules you can “feed” to the class to move them along. Ask them (a) Describe this rule in words. (b) Is it better to get a high score or a low score or some other score? (c) Which teams win for each? (Note, some of these rules produce ties).

      1. Rule 1: sum(heights-predicted.value == 0) words: the number of exactly correct predictions

      2. Rule 2: sum(heights-predicted.value) words: the sum of the differences between predicted value and the actual heights

      3. Rule 3: sum(abs(heights-estimate)) words: the sum of the absolute values of the deviations

      4. Rule 4: sum((heights-estimate)^2) words: the sum of the squared deviations

        Note: It is unlikely that students will think of the last two. That’s okay, because we will introduce them in a future lesson, but you might want to present one (or both) to see what they think about these rules.

  8. Allow student teams time to discuss and complete the task for Plot A.

  9. Do not share their responses to Plot A. Instead, display the following questions:

    1. What if we had a different set of 10 randomly selected students and plotted their heights?

    2. Would the same team win?

  10. Allow teams to discuss the questions, then share a couple of responses to the questions in the previous step.

  11. Display Plot B, found on page 2 of the Heights of Students at a High School handout (LMR_4.4), then have them find the winner using this new sample. Is it the same as they chose before?

    Note: We do NOT know the value of the true population mean/typical value. This is what we are really trying to predict.

  12. Teams will take turns to share their work as follows:

    1. Which team did you select as the winner using Plot A?

    2. Explain the method, or rule, your team used to declare the winner.

    3. Which team did you select as the winner using Plot B? Is the winner the same?

    4. Did you use the same rule to select a winner or did it change? If it changed, explain.

  13. During the share out, students will take notes about the other teams’ rules in their DS journals.

  14. Teams may continue to share at the start of the next lesson, if they run out of time.

Class Scribes:

One team of students will give a brief talk to discuss what they think the 3 most important topics of the day were.