1308 SIP Activity 2-Analysis of Coin Toss-COMPLETE

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1308

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Feb 20, 2024

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SIP Activity 2: Coin Toss Analysis Coin Tosses (Part 1) In this activity each student (groupmate) will first toss a coin 10 times (or access the StatCrunch applet* to simulate 10 tosses) and repeat the process to produce 4 sets of 10 tosses each . *StatCrunch applet is found via QUICK LINKS > MyLab StatCrunch > StatCrunch website > Interactive applets > Simulations > “Probability of a head with a fair coin.” You may select 5 flips at a time in the applet . Record your results in the table below. The exact sequence of heads and tails should be written in the Outcomes columns. Set # Outcomes (Head or Tail) No. of Heads 1 H T H H T H T T H H 6 2 H H T T H T T H H H 6 3 T H T T H T T T T T 2 4 T T H H T H H T H T 5 FIRST POST : Each groupmate must post the results of their four sets in the Discussion board. After all groupmates have posted their results, compile the data for “No. of Heads” of the group in the table below. For example , if there are 5 groupmates, there should be 20 sets; add rows as needed if you have more. Set # No. of Heads 1 6 2 6 3 2 4 5 5 6 6 5 7 3 8 5 9 4 10 5 11 5 12 6 13 7 14 5 15 3 16 4 17 6 18 3 19 4 20 5 etc… Make a histogram of the ‘Frequency of sets resulting in __95___ number of heads ,’ i.e., the frequency with which a certain number of heads occurred: 0 time, 1 time, 2 times, 3 times, … etc., among all the group’s sets. Your histogram can be created with StatCrunch or Excel and inserted into this document. Or use the blank graph provided on the next page to draw your histogram. Page 1 of 4

SECOND POST (and subsequent posts) : Share your histogram of the group’s results, and your answers to all remaining questions and tables of SIP Activity 2, in the Discussion board. Empirical Probabilities of Coin Tosses Consider the group’s relative frequencies of obtaining a certain number of heads as empirical probabilities. So, based on all the group’s sets , calculate the following empirical probabilities. 1. What is the empirical probability of obtaining 0 heads in 10 tosses? 0% 2. What is the empirical probability of obtaining 1 head in 10 tosses? 0% 3. What is the empirical probability of obtaining 9 heads in 10 tosses? 0% 4. What is the empirical probability of obtaining 10 heads in 10 tosses? 0% 5. Based on your group’s empirical probabilities, were any of the events, 0, 1, 9, or 10 heads, considered unusual events? Why or why not? Yes, 0, 1, 9, and 10 are considered empirically unusual. As a group, we had zero outcomes of either of those values. Any event that is < 0.05 or 5% is considered unusual, and each of these four values in question meet the criteria. Page 2 of 4

Theoretical Probabilities of Coin Tosses (Part 2) Think of tossing a coin 10 times as a binomial probability experiment, where the probability of heads is p = 0.5 . Compute the remaining theoretical probabilities of obtaining a certain number of heads in 10 tosses in the table below.* Enter your earlier empirical results in the last two columns. ( Ex : the empirical probability of obtaining exactly 3 heads is the proportion of all your group’s sets that resulted in obtaining exactly 3 heads.) No. of heads No. of combinations Probability of an individual outcome Theoretical Probability of Event, “No. of heads.” Frequency of sets with this No. of heads Empirical Probability of this No. of heads 0 10 ! 0 ! ( 10 − 0 ) ! = 1 1 2 1 × ( 1 2 ) 0 × ( 1 − 1 2 ) 10 = 0.001 0 0.0 1 10 ! 1 ! ( 10 − 1 ) ! = 10 1 2 10 × ( 1 2 ) 1 × ( 1 − 1 2 ) 9 = 0.01 0 0.0 2 10 ! 2 ! ( 10 − 2 ) ! = 45 1 2 45 ( 1 2 ) 2 ( 1 − 1 2 ) 8 = 0.044 1 0.05 3 10 ! 3 ! ( 10 − 3 ) ! = 120 1 2 120 ( 1 2 ) 3 ( 1 − 1 2 ) 7 = 0.117 3 0.15 4 10 ! 4 ! ( 10 − 4 ) ! = 210 1 2 210 ( 1 2 ) 4 ( 1 − 1 2 ) 6 = 0.205 3 0.15 5 10 ! 5 ! ( 10 − 5 ) ! = 252 1 2 252 ( 1 2 ) 5 ( 1 − 1 2 ) 5 = 0.246 7 0.35 6 10 ! 6 ! ( 10 − 6 ) ! = 210 1 2 210 ( 1 2 ) 6 ( 1 − 1 2 ) 4 = 0.205 5 0.25 7 10 ! 7 ! ( 10 − 7 ) ! = 120 1 2 120 ( 1 2 ) 7 ( 1 − 1 2 ) 3 = 0.117 1 0.05 8 10 ! 8 ! ( 10 − 8 ) ! = 45 1 2 45 ( 1 2 ) 8 ( 1 − 1 2 ) 2 = 0.044 0 0.0 9 10 ! 9 ! ( 10 − 9 ) ! = 10 1 2 10 ( 1 2 ) 9 ( 1 − 1 2 ) 1 = 0.010 0 0.0 10 10 ! 10 ! ( 10 − 10 ) ! = 1 1 2 1 ( 1 2 ) 10 ( 1 − 1 2 ) 0 = 0.001 0 0.0 *Hint: You may use StatCrunch > Stat > Calculators > Binomial to quickly calculate the Theoretical Probabilities. Page 3 of 4

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Theoretical vs. Empirical The theoretical probabilities of obtaining a certain number of heads in 10 tosses of a coin were calculated using the binomial probability function. 1. Would it be unusual, theoretically, to observe 0, 1, 9, or 10 heads in 10 tosses of a coin? Why or why not? (Hint: What are the probabilities of these events?) It would be unusual to observe a 0, 1, 9, or 10 because the probabilities of each are 0.001, 0.01, 0.01, and 0.001 respectively. Any event that is < 0.05 or 5% is considered unusual, and each of these four values in question meet the criteria. 2. Discuss the differences, if any, between the theoretical probabilities in part 2 and your empirical probabilities in part 1. Do differences indicate that the coins were not fair coins? What may be done to narrow the gap between theoretical and empirical probabilities? Differences between theoretical and empirical probabilities do not define the fairness of the coin. Theoretical probabilities are just that, theories. In order to close the gap between theoretical and empirical probabilities, the number of coin tosses must be a much larger number. Page 4 of 4