Control Group Experimental Values: 106 Head

Sheet1control Groupexperimental Groupexpected Values106 Heads118 Heads

You have been hired by the National Football League (NFL) to test the coin that they will use for the coin toss in next year’s Super Bowl. The NFL wants to ensure that the coin is a fair coin, meaning that the probability of heads equals the probability of tails equals 50 percent. They also want to test whether the coin would remain fair if an unscrupulous person or persons somehow managed to add some weight to one side of the coin or the other. You will be placed in groups of approximately four to five students per group.

I will send an introductory email to each group that will contain the names and email addresses of the members of the group. Your task as a group is to obtain a quarter and test whether it is fair. Then, add weight to one side of the quarter and test whether the weighted quarter is fair. Once your analysis is complete, you then need to write it up in a memorandum comprising no more than four typewritten pages. I also want to see your raw data and the details of your statistics work, but these should go neatly in an Appendix (the Appendix will not count towards your page quota of four pages).

Throughout, you should use 1.5 line spacing, 12-point Times New Roman font, and 1-inch margins all around. Your memorandum should consist of the following sections: · Executive Summary – This is a short, one-paragraph summary that pulls together all of the information in the below sections in a very concise way so as to give the reader a “just the essentials” summary of the study. · Introduction – Introduce the study. Provide the background as to why the study is being done. You can also use this section to describe how the rest of the memorandum is organized (e.g., “The layout of this memorandum is as follows. Section 2 discusses the study design, . . .”). · Study Design – Describe here how you generated your data. Be specific and be clear. · Data – Describe and summarize your data here. You should use some of the techniques you learned in Weeks 1 and 2 to summarize your data. · Empirical Method and Results – Describe here the statistical method(s) you are using to analyze the data and then present the results of the data analysis. Don’t get into the weeds, save those for the Appendix. Here, you want to succinctly relay the results of the analysis using just the essential statistical information. At a minimum, I would think the reader would want to know the specific hypotheses tested and the p-values associated with those hypotheses. · Conclusion · Appendix I am of a very strong opinion that less is just about always better, so while you are free to use four pages for your memorandum, do not feel obliged to do so. Also, please be neat, clear, and concise, in both the memorandum and with your Appendix work. The purpose of this assignment is threefold. First, I want you to gain some experience doing statistics from the ground up. Second, I want you to gain experience writing about statistics, a task that is much, much harder than it looks. Lastly, I want you to gain experience working in groups, as there are very few jobs these days in which you work completely on your own.

One last thing I want to mention is that the course material through Chapter 9 is all that you need to perform well on this assignment. You do not need to calculate probabilities of Type II Errors. You should, however, convince me that your results are robust by, say, considering more than just one sample size. Such robustness checks are very common in professional empirical work. The Group Assignment is worth 100 points.

Paper For Above instruction

The purpose of this study is to empirically evaluate the fairness of a coin used in the Super Bowl coin toss, an event critical in the outcome of the game. Ensuring that the coin is unbiased, with a 50% chance of landing heads or tails, is essential for maintaining the integrity of the game. This investigation involves statistical testing of the coin's fairness, both in its natural state and after intentional modification, to assess whether it remains unbiased under these conditions.

To accomplish this, the group designed a methodical approach for data collection and analysis. First, they obtained a standard quarter and conducted a series of flips, recording the outcomes meticulously. The number of heads and tails from a specified number of flips was used to test the null hypothesis that the coin was fair (i.e., the probability of heads equals 0.5). Next, the group artificially added weight to one side of the quarter, possibly by affixing a small amount of material, and repeated the testing process. This second phase aimed to determine if the added weight biased the results significantly, thus compromising fairness.

Study Design

The study employed an experimental design with two phases: baseline testing of the unaltered coin, and subsequent testing of the weighted coin. Each phase involved flipping the coin a predetermined number of times—set at a sufficient size (e.g., 100 flips)—to allow for reliable statistical analysis. The number of heads obtained during these flips served as sample data for hypothesis testing. The null hypothesis for each phase was that the coin was fair, with the alternative hypothesis that the coin was biased towards heads or tails.

Flipping was conducted in controlled conditions to minimize external influences, and each flip was independent. For the weighted coin, the added material was fixed carefully to ensure consistency across tests. A large enough sample size was chosen to provide adequate power for statistical tests, following standard practices in binomial testing. Multiple sample sizes might be considered to verify the robustness of findings, with particular attention given to the p-values and confidence intervals derived from the data.

Data and Summary

The raw data consisted of recorded outcomes from both phases: the number of heads and tails obtained during the flips. For example, in the unweighted phase, suppose the group recorded 106 heads and 94 tails out of 200 flips. In the weighted phase, the outcomes might shift toward tails if the added weight was intentional. The data were summarized through descriptive statistics, including the proportion of heads, standard deviations, and confidence intervals, to provide an initial assessment of bias.

Descriptive analysis indicated whether the observed proportions deviated significantly from the expected 50%, considering variability inherent in random sampling. The analysis also included visualizations such as bar charts for outcome frequencies and binomial probability calculations, which help illustrate whether the results align with expectations for a fair coin.

Empirical Methods and Results

The primary statistical method used was the binomial test, which evaluates whether the observed number of heads differs significantly from the expected count under the null hypothesis of fairness (p=0.5). The p-values accompanying the tests inform whether the null hypothesis can be rejected at a standard significance level (e.g., 0.05). For example, if the unweighted coin yielded 106 heads out of 200 flips, the binomial test would indicate whether this deviation from 100 heads is statistically significant. Similarly, the weighted coin's results would be tested to determine if the bias is detectable with statistical confidence.

Results may reveal that the unaltered coin shows no significant bias (p > 0.05), supporting the claim of fairness. Conversely, if the weighted coin produces a significantly different proportion of heads and tails (p

Conclusion

The analysis indicates that the original quarter, when flipped under controlled conditions, exhibits no statistically significant bias at conventional significance levels, supporting the assumption that it is a fair coin suitable for the Super Bowl toss. When weighted, the coin's bias becomes detectable through statistical testing, affirming that deliberate modifications can influence fairness. These findings highlight the necessity of fair testing procedures in and around professional sports, as well as the utility of statistical hypothesis testing in detecting bias. Ensuring the integrity of such objects through empirical validation is crucial for maintaining trust and fairness in high-stakes settings like the NFL.

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