Lesson 13: The Confidence Game

Objective:

Students will learn about informal confidence intervals for making estimates about population parameters based on statistics from random samples.

Materials:

The Confidence Game handout (LMR_3.13_Confidence Game)
Dotplot titled “Number Correct” displayed on the board (or on poster paper)

Vocabulary:

inferences interval confidence interval

Essential Concepts:

There is uncertainty when we estimate population parameters. Because of this, it is better to give a range of plausible values, rather than a single value.

Lesson:

Remind students that they have been learning about why sampling allows us to make inferences about a population. Some methods of sampling produce biased sample statistics, which does not allow us to generalize the results from a sample to the population of interest. To obtain unbiased statistics, random sampling methods need to be used.
Conduct a class brainstorm about what it means to “estimate” something. Have students come up with possible synonyms for the word “estimate.” Some example synonyms include: guess, approximation, projection, opinion, impression, etc.
Inform students that, in statistics, to provide an estimate means that we can give a range of values that we are confident include the population parameter value.
In today’s lesson, explain that the students will be playing a game, called The Confidence Game, in which they will be asked a series of questions that each have one numerical answer. However, instead of guessing what the exact answer is, the students will create a range of possible values that they think might include the real answer. They should be 90% confident that the true value is within their interval.
Introduce The Confidence Game to students by first going through an example using the question:

How tall is the Empire State Building, in feet (including the spire at the very top)?
1. Ask the students to write down an interval, or range, of values that they think contains the true height of the building.
2. Have a few students share their intervals with the class and discuss any obvious similarities or differences between them.
  
  For example: If Student A gives an interval from 500 to 2000 feet and Student B gives an interval from 1100 to 1400 feet, one discussion could stem from asking Student A why he or she isn’t as sure of the answer as Student B is (since Student B gave a narrower interval). Then see if Student A wants to change his or her interval.
3. After the discussion, tell the students that the actual height of the Empire State Building is 1,454 feet tall. Take a poll to see how many students’ intervals contained this value. We will learn what it means to have the true value in our intervals after we play the game.
Now, we can actually play the game! Distribute The Confidence Game handout (LMR_3.13) and explain the rules. Students will have about 5 minutes to complete the handout, which gives them approximately 30 seconds per question.

Note: The rules are printed at the beginning of the handout. They are included here for your convenience.
1. Each question must be answered WITH AN INTERVAL.
2. You should choose your interval so that you are “90% confident” (whatever that means to you).
3. You CANNOT use any reference tools (i.e. no cell phones or computers to find answers).
4. A question is “correct” if the true answer is inside your interval.
5. The winner is determined by who got the most questions correct. In the case of a tie, the winner is chosen by whose intervals were narrower.
LMR_3.13
Once each student has completed The Confidence Game handout (LMR_3.13), have students choose partners and exchange handouts so that they can grade each other. Remind them that a question is marked as “correct” if the actual value (see answers in Step 8) falls within the interval.
Display the answers for each of the 10 questions from the handout found below:

1) In what year did Mickey Mouse make his film debut? 1928

2) What is the lowest temperature (in degrees Fahrenheit) ever recorded in California? -45 degrees Fahrenheit

3) During the year 2014, how many television series were aired? 1,715 TV shows

4) How far away, in miles, is Earth from the moon? 238,900 miles

5) What is the greatest number of children officially recorded that were all born to one mother? 69 children

6) In what year did Orville and Wilbur Wright, more commonly known as the Wright brothers, make the first-ever powered flight in a plane? 1903

7) As of June 2015, how many of Rihanna’s songs have reached the Number 1 spot on Billboard’s “Dance Club Hits” chart? 23 songs

8) How many years have actors Will Smith and Jada Pinkett-Smith been married? As of January 2016, it was 18 years - they were married December 31, 1997.

9) How many hours will it take to complete a cross-country road trip from Los Angeles to New York City according to Google Maps? 41 hours (2,789.5 miles)

10) How many baseball fans can attend game at Dodger Stadium during any given day? 56,000 fans
Each student should write the total number of “correct” responses at the top of his or her partner’s handout, and then return it.
Engage the students in a discussion about how well they did at estimating the true values with their intervals. The following questions can be used to steer the discussion:
1. Remember that we were aiming to be 90% confident for each question. Based on this, how many of the 10 questions should we each have gotten correct? If we are 90% confident, then we would expect 90% of the 10 intervals to include the true value, which is 9 intervals.
2. Did anyone in the class get exactly 9 correct? Did anyone get all 10 correct? Answers will vary by class. However, it is very unlikely that many students will have gotten 9 or 10 correct responses on this first round.
Create a dotplot on the board (or on poster paper) titled “Number Correct” and have each student record his or her value. Then, ask:
1. How many students got 9 correct? In other words, how many students were actually 90% confident of their intervals? Answers will vary by class.
2. What is the typical number of correct responses for our class? Does it seem too high or too low? Explain. Answers will vary by class. Most likely, the typical number of correct responses will be fairly low (maybe even 4 or less).
3. Why is our typical score so much lower than 9? We tend to be more confident than we should be, so we create narrower intervals.
4. It looks like, even though we thought we were 90% confident, most of us (or all of us) did not succeed 90% of the time. How could we increase our level of confidence? We could use wider intervals.
Recall from Step 3 that, in statistics, to estimate something means that we can give a range of values that we are confident include the population parameter value. This range of values, like the ones the students created during The Confidence Game activity, is known as a confidence interval.
Students will continue to learn about confidence intervals during the next lesson.

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.

Homework

In your own words, write a description of what a confidence interval is and why it is used in statistics.