Statistics Activities for a High School Mathematics Class Room

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Statistics Activities for a High School
Mathematics Class Room

• Megan McLennan
• May 2, 2005

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Outline

• Introduction
• Activities
• The Price is Right
• Master Key, Plinko
• Probability of a Kiss
• Activity 1, Activity 2, Activity 3
• Hypothesis Testing vs. Jury Trial
• Conclusion

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Introduction

• Why I chose this topic
• High school teacher
• Have some ideas for how to incorporate stats in a
  math classroom before I go out into the field
• Personal experiences with statistics in high
  school
• Not necessary to have a strong stats background
  to be a high school math teacher

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The Price Is Right

• The game show The Price is Right consists of
  contestants playing product-pricing games in
  order to win prizes.
• Need the knowledge of prices, but there is also
  an element of chance.
• There are roughly 70 games on TPIR, I will go
  through two of the games that can be made into a
  classroom activity for high school students
• Master Key, Plinko

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Master Key- about the game

• There are three prizes the contestant can win, a
  small prize, and medium prize and a large prize
• There are five keys randomly placed in front of
  the contestant, one for the small prize, one for
  the medium prize, one for the large prize, one
  for all the prizes, and one that is a dud
• The contestant has a chance to pick up to two
  keys.
• He is shown a product for which two prices are
  given, if he guesses the right price, he gets a
  key. This repeated again
• The contestant uses the keys he earned to try to
  open the three locks and wins whatever he unlocks

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Master Key- Goals

• Computing Probabilities
• Conditional Probabilities
• Bayes Rule
• Counting
• Combinations
• Ex. If you have 10 objects and choose 3 of them,
  how many combinations are possible?

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Master Key- Questions

• Assume that the contestant has no pricing
  knowledge of the two products, and therefore
  his/her decisions of choosing the correct price
  for each product are independent and each has a
  50 chance of being right. Compute the following
  probabilities

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Master Key- Questions

• A. What is the probability that the contestant
  wins no prizes?
• B. What is the probability that the contestant
  wins a prize, but not the large prize?
• C. What is the probability that the contestant
  wins the large prize?
• D. Given that a contestant has won the car, what
  is the probability that he had earned only one
  key?

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Master Key- Solutions

• Consider the distribution for the number of keys
  earned, X, and the conditional probabilities for
  what prizes can be won given X keys were earned.
  Let
• A win no prizes,
• B win the small and/or the medium prize but not
  the large prize,
• C wins the large prize

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Master Key- Solutions

• The distribution of X and the conditional
  probabilities of
• A, B, and C given
• X can be displayed
• in a tree diagram

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Master Key- Solutions

• Breaking down the tree
• P(A X0) 1
• P(A X1) 1/5
• P(B X1) 2/5
• P(C X1) 2/5
• P(A X2) 0
• P(B X2) 3/10
• 5 C 2 10, ways to choose the two keys
• 3 C 2 3, ways can win the small and/or medium
  prize, but no car
• P(C X2) 7/10

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Master Key- Solutions

• Using the tree (and p1 p2 0.5)
• A. P(A) (1-p1)(1-p2)(1) p1(1-p2)
  p2(1-p1)(1/5)
• P(A) 14/40
• B. P(B) p1(1-p2) p2(1-p1)(2/5) p1p2(3/10)
• P(B) 11/40
• C. P(C) p1(1-p2) p2(1-p1)(2/5) p1p2(7/10)
• P(C) 15/40

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Master Key- Solutions

• D. Given that a contestant has won the large
  prize, what is the probability that he had earned
  only one key?
• P(X1 C) can be computed using Bayes Rule (on
  side)
• P(X1 C) 8/15

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Plinko- about the game

• The contestant is given one chip and has the
  opportunity to win 4 more.
• To earn the other 4 the contestant is presented
  with products that are given a price and must
  guess if the actual price is higher or lower. For
  each correct response, a chip is rewarded.

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Plinko- about the game

• After the contestant has their chips, they must
  drop the chips from any of the nine slots at the
  top of the Plinko board.
• On its way down, the chip will encounter 12 pegs.
  If the chip hits a peg next to the wall it will
  fall into the only open slot, otherwise it will
  fall to the left or the right of the peg with
  equal probability.
• The highest prize for one chip is 10,000, so up
  to 50,000 can be won.

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Plinko- Goals

• Binomial Experiments
• Experiments consisting of the observation of a
  sequence of identical and independent trials,
  each of which can result in one of two outcomes.
• Expected Value of a Random Variable
• Counting
• Combinations of n objects taken r at a time

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Plinko- Question

• Contestants with multiple chips usually vary the
  slots from which they release the chips. Does the
  initial placement of the chip matter? To decide,
  answer the following questions

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Plinko- Question/ Answer

• Question 1 For each
• of the three middle
• slots at the top of the
• Plinko board (slots 4,5,
• and 6), find the
• probability that a
• chip starting in
• each slot results in
• winning 10,000.

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Plinko- Question/ Answer

• Answer 1
• Let Y of pegs out of 12 that result in the
  chip falling to the left. (Y is not a binomial
  random variable because of the constraints
  imposed by the walls of the board)
• For a chip dropped in slot 5 to win 10,000, the
  chip must fall to the left exactly 6 times (Y6).
  (If the chip hits a wall of the board it has
  moved to the left or right at least 8 times, and
  could not end up in the 10,000 bin. Thus, we may
  use the binomial distribution with n12 and
  p0.5)
• P( win 10,000 starting from slot 5) P(Y6)
  (12 C 6)(1/2)12 .2256
• For a chip dropped in slot 4 it will win 10,000
  only if the chip falls to the left exactly 5
  times (and to the right 7 times). Binomial
  distribution applies here as above.
• P( win 10,000 starting from slot 4) (12 C
  5)(1/2)12 .1934
• Same answer for slot 6 as slot 4.

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Plinko- Question/ Answer

• Question 2 If a chip is dropped in the middle
  slot of the Plinko board (slot 5), the amount
  won, U, has the following distribution
• If a chip is dropped in either of the slots
  adjacent to the middle slot (slot 4 or 6), the
  amount won, V, has the following distribution
• Compute the expected winnings for a chip dropped
  in slot 5 and the expected winnings for a chip
  dropped in slot 4 or 6.

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Plinko- Question/Answer

• Answer 2
• Using slot 5
• Using slot 4 or 6

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Price is Right- Conclusion

• I like this project because most people like
  TPIR.
• Lots of ways to present this activity
• For example, can start out by watching clips of
  the games to get the students excited.

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Probability of a Kiss- Activities

• Activity 1
• Collecting and analyzing data
• Activity 2
• Make predictions and displaying data
• Activity 3
• Properties of the distribution of a sample
  proportion

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Probability of a Kiss- Activity 1

• Materials
• 10 plain Hersheys Kisses
• 16-oz plastic cup
• Students should be in groups of 3 or 4

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Probability of a Kiss- Activity 1

• Procedure
• Students discuss and estimate the probability a
  Kiss will land on its base when it is tossed on
  the desk
• Leads to discussion of three types of
  probabilities, empirical, subjective, and
  theoretical.
• Empirical probability can be thought of as the
  most accurate scientific "guess" based on the
  results of experiments to collect data about an
  event.
• Subjective probability describes an individual's
  personal judgment about how likely a particular
  event is to occur.
• Theoretical probability is the ratio of the
  number of ways the event can occur to the total
  number of possibilities in the sample space.
• In this case, subjective probabilities are being
  assigned.

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Probability of a Kiss-Activity 1

• Now groups put 10 Kisses into the cup and spill
  them onto the desk and record the number of
  Kisses that have landed on their base in a table,
  including a row for the total number of base
  landings. (Repeat 10 times).
• After the groups are done they are asked to
  refine their previous guesses of the probability
  the Kiss will land on its base.
• The students engage in a class discussion where
  they are asked how they could be more certain of
  the probabilities. (More tosses necessary)

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Probability of a Kiss- Activity 2

• Materials
• In addition to the 10 plain Kisses, also 10
  almond Kisses.
• 16-oz plastic cup
• Students in groups of 3 or 4 again

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Probability of a Kiss- Activity 2

• Procedure
• Students compare the two types of Kisses and
  discuss which would have the higher probability
  of landing on its base.
• Students then put all 20 Kisses into the cup and
  spill them on the table and record the number of
  Kisses that have landed on their base for each
  type. Repeat 10 times.
• Students learn how to deal with messy data

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Probability of a Kiss- Activity 2

• Displaying Data
• Stem and Leaf Plots
• Ex Plain Almond
• 1 8 9
• 8 8 4 2 0 2 4 6
  6 7 8 8 8
• 7 6 6 5 4 2 2 1 0 3 0 0 1 3 3 3
  6
• 3 3 0 4

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Probability of a Kiss- Activity 2

• Displaying data
• Boxplots
• Ex
• Reviews finding 1st and 3rd quartiles, medians,
  max and mins
• Students should find outliers for both data sets
• Calculate means and standard deviations

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Probability of a Kiss- Activity 3

• Materials
• 30 plain Kisses, 30 almond Kisses, 16-oz cup
• Procedure
• Groups spill 10 plain Kisses onto the desk and
  record the number that land on its base (repeat 5
  times)
• Repeat using 20 plain Kisses
• Repeat using 30 Kisses
• Do the same for the almond Kisses

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Probability of a Kiss- Activity 3

• Procedure (cont.)
• Groups combine results with a partner group to
  obtain five tosses for n60 and n90 for each
  type of Kiss.
• Record proportions from both plain and almond
  Kisses in a table.
• Calculate standard deviation and mean for the
  sample proportions and interpret.
• Which sample size has a larger standard
  deviation? Why? (Analyze plain and almond tables
  separately.)

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Probability of a Kiss- conclusion

• This is a good activity for students to develop
  critical thinking skills, with the class
  discussions.
• Also allows students to display their own data
  findings in different ways

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Hypothesis Testing

• Hypothesis Testing can be confusing for students
  to understand. It is important for students to
  understand this concept.
• Teaching hypothesis testing using a jury trial as
  an example.

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Hypothesis Testing

• Put the students in groups of 3 or 4
• Each group gets 12 note cards, each note card has
  one of the following phrases
• Parts of Hypothesis Testing
• Null Hypothesis, Ho
• Alternative Hypothesis, Ha
• Test Statistic
• Rejection Region
• Decision
• Conclusion

• Parts of a Jury Trial
• Original Claim person presumed innocent
• Want to prove person is guilty
• Court Case evidence presented
• Judges words on the case
• Jury Deliberations
• Verdict

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Hypothesis Testing

• First the group goes through and defines the
  phrases on the hypothesis testing note cards and
  writes the definitions on the back of the note
  card.
• After that they must match the parts of the
  hypothesis testing note cards to the
  corresponding jury trial cards.
• When all groups are finished, the class
  reconvenes and discuss their answers.

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Hypothesis Testing

• Now each group gets 4 more note cards
• Type I error
• Type II error
• Innocent person found guilty
• Guilty person found innocent
• Each group then must define Type I and Type II
  errors in the context of hypothesis testing on
  the back of the card and then match those note
  cards to the corresponding jury trial note cards.

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Hypothesis Testing

• Type I and Type II Errors
• Type I Error
• Ho is true, but Ha was concluded.
• Innocent person was found guilty.
• Type II Error
• Ho is false, but Ha was not concluded.
• Guilty person was found not guilty.

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Hypothesis Testing- conclusion

• I like this activity because it puts the process
  of hypothesis testing into a real-life scenario
  students can understand and are familiar with.
• Also, the students are forced to review the
  definitions involved with hypothesis testing.

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Helpful Links

• Links to web-based interactive statistics
  activities
• ESP activity
• Binomial Experiment
• Let's Make a Deal
• Empirical Techniques, repeated trials

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Conclusion

• Making sure that high school students understand
  statistics is very important
• High school kids are usually turned off by
  numbers and they need to be presented with new
  concepts in ways that keep them interested.

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Thank You

• Jong- Min Kim, advisor
• Audience and Friends

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References

• Wackerly, Dennis D., William Mendenhall III,
  Richard L. Scheaffer. Mathematical Statistics
  with Applications. Pacific Grove Duxbury, 2002.
• Biesterfeld, Amy. Journal of Statistics
  Education. The Price (or Probability) is Right.
  Volume 9, Number 3 (2001). University of Colorado
  at Boulder. e/v9n3/biesterfeld.html.
• Richardson, Mary, Susan Haller. Journal of
  Statistics Education. What is the Probability of
  a Kiss?. Volume 10, Number 3(2002), http//www.amstat.org/publications/jse/v10n3/halle
  r.html.
• McCullough, Desiree A., Jury Approach to
  Hypothesis Testing. September 27-28, 2002.
  University of Tennessee at Martin.
  les/frame.htm.
• Sungur, Engin. EXTRASENSORY PERCEPTION (ESP).
  University of Minnesota, Morris.
  lic/instruction/esp/esp.shtml.
• West, R. Webster. Lets Make a Deal Applet.
  University of South Carolina. edu/west/javahtml/LetsMakeaDeal.html.

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Questions