{"id":6112,"date":"2015-09-26T09:35:40","date_gmt":"2015-09-26T04:05:40","guid":{"rendered":"http:\/\/ucanalytics.com\/blogs\/?p=6112"},"modified":"2016-09-14T12:50:46","modified_gmt":"2016-09-14T07:20:46","slug":"data-analytics-challenge-1-solve-the-case-of-a-shady-gambler-clue-2","status":"publish","type":"post","link":"https:\/\/ucanalytics.com\/blogs\/data-analytics-challenge-1-solve-the-case-of-a-shady-gambler-clue-2\/","title":{"rendered":"Data Analytics Challenge 1(Clue # 2) : Solve the Case of a Shady Gambler"},"content":{"rendered":"
\"Sherlock<\/a>

Data Analytics Challenge \u2013 by Roopam<\/p><\/div>\n

\n

Sometimes answers lead to more questions. In data science<\/span> this sometime is true almost everytime.<\/p><\/blockquote>\n

Analytics Challenge – The Shady Gamble<\/span><\/h2>\n

In the previous article<\/a>, you were approached by Scotland Yard to investigate charges against a gambler with dubious character. The allegation against him was that he had loaded his die so that some numbers are more likely to appear after a throw than others. They were seeking your help to understand whether the gambler\u2019s die is biased\u00a0or not? They had shared the following barplot representation of the gambler’s previous 1000 throws.<\/p>\n\n\n\n
As mentioned in the previous part, all the analytics\u00a0challenges on YOU CANalytics require your participation to proceed. We have some good discussion running in the comments section of the previous part (discussion link<\/a>). This part will reveal a few more clues for you to further that discussion.\u00a0I will use your comments from the previous part, along with some creative freedom, to move ahead with\u00a0this investigation.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\"Mystery<\/a><\/p>\n

Your\u00a0initial reaction after looking at this plot was : the gambler’s\u00a0die looks biased. You\u00a0know that for a fair die each number on the die has the equal\u00a0likelihood of appearing. In other words, the length of all the bars in the above bar chart must be roughly the same. Since that doesn’t seem to be the case hence the die must be biased.<\/p>\n

Later after a more careful analysis, you noticed that the bar plot is zoomed on the Y-axis. In this case,\u00a0Scotland Yard has used Sherlock Holmes’\u00a0favorite instrument for detection i.e. magnifying glasses.<\/p>\n

Realizing this, you make\u00a0the appropriately-scaled version of the same plot (look this one starts from 0 on the Y-axis).<\/p>\n

\"Slide4\"<\/a><\/p>\n

Please help Scotland Yard answer the following questions. Like the last time please add your comments in the discussion section at the bottom of this article.<\/p>\n\n\n\n\n\n
1<\/strong><\/td>\nDo you still feel that the die is biased?\u00a0Why or why not?<\/td>\n<\/tr>\n
2<\/strong><\/td>\nIs the use of magnifying glass (Sherlock Holmes’ favorite instrument for detection) justified here? Why or why not?<\/td>\n<\/tr>\n
3<\/strong><\/td>\nCould you see some similarity between this analysis and the way the popular press reports economic indicators (stock market in particular)?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

To continue with your analysis, you have placed the bars of expected values i.e. 1000\/6 = 166.67 next to the observed value for the gambler’s die. The following is the same plot.<\/p>\n

\"Slide6\"<\/a><\/p>\n

Now, our original question can be rephrased to: are the blue bars and the orange bars significantly<\/span> different from each other? If yes then the coin is biased. At first, to answer this question you had suggested simulation. The idea behind\u00a0simulation is to repeat the experiment of rolling 1000 trials of a die several times through a random number generator and compare those results with our observation.<\/p>\n

Simulation is a good way to test our hypothesis however for this problem we will employ a much more theoretically robust method i.e. Chi-Square goodness-of-fit test. In it’s simplest form Chi-Square GOF is a normalized difference between expected and observed distributions, and comparing this difference against a standard value (i.e. Chi-Square distribution). Scotland Yard wants to know:<\/p>\n\n\n\n
4<\/b><\/td>\nWhat is the difference between simulation and Chi-square test? How are they similar to test random events?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

You\u00a0wrote down the following lines of R code to calculate Chi-Square statistics\u00a0(Credit for this code: Jason Corderoy \/ Ram)<\/span>.\u00a0<\/em>The first line of the code takes in observed value of frequency for each number on the die<\/p>\n

Observed<\/span> <- c<\/span>(162,160,156,173,182,167)<\/pre>\n
The next line of the code produced expected frequency through uniform probability distribution where each number on the die has 1\/6 chances of appearance.<\/p>\n
Expected<\/span> <- rep<\/span>(1\/6,6)<\/pre>\n
The subsequent line of the code does the Chi-Square goodness of fit test for observed and expected values.<\/p>\n
test.chi<\/span> <- chisq.test<\/span>(Observed<\/span>,p=Expected<\/span>)<\/pre>\n
The final line of the code test the hypothesis at 0.05 significance level i.e. 5% chances that observed values are not the same as expected values.<\/p>\n
ifelse<\/span>(test.chi<\/span>$p.value <= 0.05,\r\n       \"NULL Hypothesis Rejected: Die not fair.<\/span>\",\r\n       \"NULL Hypothesis not rejected: Die fair.<\/span>\")<\/pre>\n
After running the last line of code, R spews out the following result: “NULL Hypothesis is not rejected: Die fair.”. The inspector from Scotland Yard is completely confused by now. He phrased the following questions:<\/p>\n\n\n\n\n
5<\/strong><\/td>\nWhat is the rationale behind choosing p-value of 0.05 as the threshold to accept or reject the hypothesis?<\/td>\n<\/tr>\n
6<\/strong><\/td>\nIs\u00a0there a better way to represent the results rather than conclusive acceptance or rejection of hypothesis? How will it help?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
 <\/p>\n
Finally, Scotland Yard provides you with one final piece of information around\u00a0values that are in favor of the house. The house always makes money when a number above 3 appears on the die\u00a0(Credit:\u00a0Terry Taerum)<\/span>. <\/span><\/span><\/em>The final question you hear from the inspector from Scotland Yard is:<\/p>\n\n\n\n
7<\/strong><\/td>\nIn the light of this new evidence i.e number 4,5 and 6 always make money for the house, would you change your conclusion about the die being fair? Why or why not?<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
 <\/p>\n\n\n\n
This time, Scotland Yard has way too many questions for you (7 to be precise). You don’t need to discuss all of them, take your pick and start a discussion in the comment section at the bottom. Please mention the question number(s) in your comment.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
 <\/p>\n","protected":false},"excerpt":{"rendered":"
Sometimes answers lead to more questions. In data science this sometime is true almost everytime. Analytics Challenge – The Shady Gamble In the previous article, you were approached by Scotland Yard to investigate charges against a gambler with dubious character. The allegation against him was that he had loaded his die so that some numbers<\/p>\n
Read More…<\/a><\/p>\n","protected":false},"author":1,"featured_media":6043,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_newsletter_tier_id":0,"jetpack_publicize_message":"","jetpack_is_tweetstorm":false,"jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[75],"tags":[],"jetpack_publicize_connections":[],"yoast_head":"\nData Analytics Challenge 1(Clue # 2) : Solve the Case of a Shady Gambler – YOU CANalytics |<\/title>\n<meta name=\"description\" content=\"Welcome back to Data Analytics Challenge on YOU CANalytics\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/ucanalytics.com\/blogs\/data-analytics-challenge-1-solve-the-case-of-a-shady-gambler-clue-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Data Analytics Challenge 1(Clue # 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1) : Solve the Case of a Shady Gambler","author":"Roopam Upadhyay","date":false,"format":false,"excerpt":"Data! data! data!\" he cried impatiently. \"I can't make bricks without clay. - Sherlock Holmes In this article, we are starting a new series on YOU CANalytics that can't proceed without your\u00a0participation to solve the data analytics mysteries. In the beginning of each article of this series, you will get\u2026","rel":"","context":"In "Analytics Challenge"","block_context":{"text":"Analytics Challenge","link":"https:\/\/ucanalytics.com\/blogs\/category\/analytics-challenge\/"},"img":{"alt_text":"Mystery Challenge plot","src":"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2015\/09\/Mystery-Challenge-plot.jpg?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2015\/09\/Mystery-Challenge-plot.jpg?resize=350%2C200 1x, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2015\/09\/Mystery-Challenge-plot.jpg?resize=525%2C300 1.5x, https:\/\/i0.wp.com\/ucanalytics.com\/blogs\/wp-content\/uploads\/2015\/09\/Mystery-Challenge-plot.jpg?resize=700%2C400 2x"},"classes":[]},{"id":6693,"url":"https:\/\/ucanalytics.com\/blogs\/data-analytics-challenge-1-clue-3-does-bayes-theorem-have-a-better-answer\/","url_meta":{"origin":6112,"position":1},"title":"Data Analytics Challenge 1 (Clue # 3) : Does Bayes’ Theorem have a Better Answer?","author":"Roopam Upadhyay","date":false,"format":false,"excerpt":"Data Analytics Challenge 1 \u2013 The Shady Gambler This is the final part of the first\u00a0data analytics challenge on YOU CANalytics. 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