Statistics Activities for a High School Mathematics Class Room Megan McLennan May 2, 2005
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
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
“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
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
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?
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:
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?
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
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:
Master Key- Solutions • Breaking down the tree: • P(A | X=0) = 1 • P(A | X=1) = 1/5 • P(B | X=1) = 2/5 • P(C | X=1) = 2/5 • P(A | X=2) = 0 • P(B | X=2) = 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 | X=2) = 7/10
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
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(X=1 | C) can be computed using Bayes Rule (on side): • P(X=1 | C) = 8/15
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.
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.
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
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:
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.
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 (Y=6). (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 n=12 and p=0.5) • P( win $10,000 starting from slot 5) = P(Y=6) = (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.
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.
Plinko- Question/Answer • Answer 2: • Using slot 5: • Using slot 4 or 6:
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.
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
Probability of a Kiss- Activity 1 • Materials • 10 plain Hershey’s Kisses • 16-oz plastic cup • Students should be in groups of 3 or 4
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.
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)
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
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”
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 |
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
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
Probability of a Kiss- Activity 3 • Procedure (cont.) • Groups combine results with a partner group to obtain five tosses for n=60 and n=90 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.)
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
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.
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 Hypothesis Testing
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.
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.
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.
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.
Helpful Links • Links to web-based interactive statistics activities • ESP activity • Binomial Experiment • Let's Make a Deal • Empirical Techniques, repeated trials
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.
Thank You • Jong- Min Kim, advisor • Audience and Friends
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. <http://www.amstat.org/publications/jse/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/haller.html>. • McCullough, Desiree A., Jury Approach to Hypothesis Testing. September 27-28, 2002. University of Tennessee at Martin. <http://www.utm.edu/staff/desireem/origtmtapres_files/frame.htm>. • Sungur, Engin. EXTRASENSORY PERCEPTION (ESP). University of Minnesota, Morris. <http://www.morris.umn.edu/~sungurea/introstat/public/instruction/esp/esp.shtml>. • West, R. Webster. Let’s Make a Deal Applet. University of South Carolina. <http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html>.