# Statistical Inference and Central Limit Theorem Project

### Description

Performance Assessment Task 4: Paper, Rock, Scissors Purpose: The purpose of this assignment is to give you the opportunity to explore whether your class displays a particular pattern when playing this game. By comparing the class results to expected outcomes obtained through simulation and applying the Central Limit Theorem, you will determine how surprising the class result is. PAT #4 Standard: I can test whether the Central Limit Theorem applies to a particular event, use the analysis from the Central Limit Theorem to draw conclusions about a claim, and understand the relationship between sample size and statistical significance. Description: The game of Rock-Paper-Scissors (or rochambeau) has been played as a selection game (like flipping a coin) for over 100 years at least.

Players count to three and then reveal one of the three hand gestures. The objective is to display a hand gesture that defeats that of your opponent. The World RPS Society Official Abridged Rules of Play can be found at www.worldrps.com, but the key idea is that rocks break scissors, scissors cut paper, and paper covers rock. It has been conjectured that there are certain patterns of play, especially by novices. The previous week, folks in your class took the Paper, Rock, Scissors Survey. We will be using these results to complete this PAT. You may complete the assignment in fours, trios, pairs or individually. Assignment Deliverable Each group will submit a Rock-Paper-Scissors Report (see following pages). Make sure you list the names of all group members.

Name(s) :__________________ Rock, Paper, Scissors Report In filling out this report, you will use the class results from playing R-P-S. Wherever there are >> marks, you should begin typing your response. You can copy and paste from this list of symbols: π, 𝑝̂ , μ, 𝑥̅ , σ, s. Scenario: Rock-Paper-Scissors is a game of chance. It is equally probable that someone will choose any one of the three options: rock, paper, or scissors. Researchers1 claim that players tend to not prefer scissors, choosing it less than 1/3 of the time. Does the researchers’ claim hold at Nazareth College? That is, do all students at Nazareth College tend to choose scissors less than 1/3 of the time? We will use our class as a representative sample of all Nazareth College students in playing R-P-S (Note: There is no reason to believe that students in MTH 207 choose different options than all Nazareth College students.) Do our sample results provide sufficient evidence to the researchers’ claim, or were the results due to chance variation?

Another way of looking at this question is to consider this: Given that Nazareth College students choose rock, paper, or scissors at random, what is the probability that the proportion of time they choose scissors is less than 1/3? Understanding the Scenario

1. What are the observational units in the scenario? >>

2. What is the variable and its classification? >>

3. How many students in our class took the R-P-S survey? What symbol do we use to represent this? >>

4. How many students picked scissors? >>

5. What proportion of students in MTH 207 picked scissors? What symbol would you use to represent this value?

>> 6. What is the population in the scenario? >>

7. What is the population parameter in the scenario?What symbol do we use to represent this value? >> Simulation Even though the context is different, you can use the Reese’s Pieces applet to conduct this simulation by doing the following using the R-P-S context located in parentheses. • • • • 1 Specify the conjectured value of π, the probability of orange (probability of choosing scissors) Use at least 3 decimal places. Specify the Number of candies of our study (the number of students). Set the Number of samples to 2000. Uncheck the Animate box. Eyler, Shalla, Doumaux, and McDevitt. (2009) Winning at rock-paper-scissors. College Mathematics Journal. • • • • Press Draw Samples. Choose the radio button Proportion of Orange. Use the As Extreme As box click the inequality so that it is less than or equal to, and type in the proportion from the sample. Press Count.Check the box Summary Stats.

8. Take a screen shot of the applet showing the histogram simulated sampling distribution, the sampling distribution mean and standard deviation, and the percentage value of samples that obtained proportions below our sample proportion and paste it below. >>

9. Based on your simulated histogram, how likely is it to obtain a sample proportion that is less than or equal to our class’ sample proportion if the population proportion equals 1/3? (Include a probability value and a likelihood designation) >> Statistical Inference – Central Limit Theorem Now let’s consider instead using the Central Limit Theorem for a Sample Proportion (as opposed to simulation) to determine the likelihood. There three “conditions” to meet to apply the CLT: • • • Randomness: We have a random sample from the population of interest Large Population: We have a population that is at least 10 times our sample size. Sample size: We have a large enough sample size, where large enough is checked by seeing whether n×π ≥ 10 and n×(1-π) ≥10.

10. Explain why our sample be considered random enough. >>

11. Show your calculations and explain as to whether we have met the condition of Large Population. >>

12. Show your calculations and explain as to whether we have met the condition of Sample Size. >>

13. What does the Central Limit Theorem predict for the shape, mean, and standard deviation of the sampling distribution of the sample proportion (assuming the null hypothesis is true)? (Show the calculation details.) >> Use this Normal model specified by the CLT, and the Normal Probability Calculator applet, to determine the probability of interest: • • • Specify an appropriate axis label for the sampling distribution we are considering in the variable box in the Normal Probability Calculator applet below, then plug in the mean and standard deviation values from the CLT(at least 3 decimal places each), and press Scale to Fit. Specify the value of our sample proportion in the X box. Make sure the symbol is < to find the area about the statistic value. Press Enter/Return on your keyboard.

14. Take a screen shot of the applet so that the shaded normal curve can be seen along with the sample proportion, the mean, and the standard deviation. Paste it below. >>

15. Provide an interpretation of the z-value in this context. >>

16. How does the estimate of the probability between the two methods (simulation versus the Central Limit Theorem) compare? Were they similar? Did you expect them to be similar based on your check of the conditions for the Central Limit Theorem? Explain. >>

17. Provide an interpretation of this probability in this context. >> Going Further Suppose our MTH 207 class had been twice as large, and you had obtained the same sample proportion choosing scissors, how would that affect the probability?

18. Using this hypothetical scenario, what are the new values for n and p̂ ? >>

19. Use either the Reese’s Pieces applet or the Normal Probability Calculator applet to obtain the probability for these hypothetical data. Include a screen capture below of your output displaying the probability value. >>

20. Did the probability change from the probability value previously? >>

21. Do these data provide stronger, weaker, or similar evidence that students at Nazareth pick scissors less than one-third of the time than what you obtained previously? >> Conclusions

22. Write a paragraph summarizing both of your analyses. Your summary should include: • How you are arriving at your decision about whether there appears to be something to the theory that people tend to choose scissors less than 1/3 of the time. Did your decisions differ between the 2 data sets (i.e. the ones with different sample sizes)? • Whether the Central Limit Theorem was applicable in this study. • The effect of the sample size on your decisions. • Was this sample randomly selected from the population of students at your school? Explain. Do you have concerns that the sample is not likely to be representative of students at your school on this topic? Explain >

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