DATA1001 – Introduction to Data Scienceand DecisionsAssignment 3Student detailsSurname:Given names:Student number:Tutorial day/time:Date due: 4pm on 26 October 2018Date submitted:DeclarationI declare that this assessment item is my own work, except where acknowledged, andacknowledge that the assessor of this item may, for the purpose of assessing this item:1. Reproduce this assessment item and provide a copy to another member of theUniversity; and/or2. Communicate a copy of this assessment item to a plagiarism checking service(which may then retain a copy of the assessment item on its database for thepurpose of future plagiarism checking)I certify that I have read and understood the University Rules in respect of StudentConduct.Signed1:Date:Mark:Comments:1Sign in ink if handing in a hard copy; or type your name if handing in an electronic copy.1A bit of Differential PrivacyThe Australian Taxation Office would like to find out the proportion of Australians whocheat on their tax returns. They select a random sample (e.g. by tax file numbers) ofAustralians and ask them if they have ever cheated on their tax return. Of course, theypromise that there will be no repercussion if the answer is ‘yes’, and that they will notrecord this information against your identity.If you had cheated on a tax return previously, would you answer truthfully?Your answer is probably ‘No’, because you wouldn’t trust the ATO to securely managethis information on you. In this assignment, you’ll explore a way in which an individualcan safely disclose this information.Procedure: The respondent secretly flips a coin twice. If the first flip shows ‘heads’,they answer truthfully. Otherwise, they answer ‘yes’ or ‘no’ according to thesecond flip being ‘heads’ or ‘tails’.The idea is that this way, enough ‘randomness’ is added to the respondents answer sothat they cannot be identified.Below, the (unknown) true proportion of tax return frauds is θ ∈ [0, 1] the variable fraud with values 0, 1 denotes whether a respondent is a fraud the variable truth with values 0, 1 denotes whether a respondent answers truthfully the variable yes with values 0, 1 denotes whether a respondent answers ‘yes’according to the above procedure.Question 1) The use for the ATOa) [3 marks]Draw a tree diagram for the three variables fraud, truth and yes.b) [3 marks]Fill in the remaining values in this probability table:fraud truth yes P0 0 0 0 0 1 1θ40 1 0 0 1 1 01 0 0 2fraud truth yes P1 0 1 1 1 0 1 1 1 Hint: according to your tree diagram,P((fraud = 0) ∩ (truth = 0) ∩ (yes = 1))= P((fraud = 0))P(truth = 0|fraud = 0)P(yes = 1|(truth = 0) ∩ (fraud = 0))= (1 ? θ)(1/2)(1/2) = (1 ? θ)/4c) [2 mark]Show that P(yes = 1) = 1/4 + θ/2.Hint: Follow all paths in your tree diagram that lead to yes=1 and add their probabilities.d) [1 mark]The ATO has received 10384 responses, of which 3448 were yes=1. From these numbers,derive an estimate of θ.Question 2) Is it really safe to participate?In this question, we’ll calculate the probability P(fraud = 1|yes = 1). If this probabilityis close to 1, then by playing the game and answering yes you will reveal yourself as atax fraud!a) [2 mark]Compute all probabilities P((fraud = i) ∩ (yes = j)) for all possible values of i andj, and collect them in a table. (This procedure is called computing the “marginaldistribution” of fraud and yes.)Hint: the event A = (fraud = 1) ∩ (yes = 0) is the disjoint union of A ∩ (truth = 0)and A ∩ (truth = 1). Find the probabilities for these events in your table.b) [2 mark]Calculate P(fraud = 1|yes = 1) and P(fraud = 1|yes = 0)3c) [2 mark]What would happen if a biased coin was used in the first toss? (Limit your answer to 3normal-length sentences.)Assignment SubmissionDue Date: 4pm on 26 October 2018 (That’s Friday in Week 13.)Hand in your assignment to the School Office of the School of Mathematics & Statistics,Level 3, Red Centre Building (Centre Wing).Late submissions20% (3 marks) will be deducted at 0, 24, 48, 72, 96 hours after the deadline. Worksubmitted more than five days late will not be marked.On PlagiarismThe University regards plagiarism as a form of academic misconduct, andhas very strict rules regarding plagiarism. For UNSW policies, penalties,and information to help you avoid plagiarism see: https://student.unsw.edu.au/plagiarism as well as the guidelines in the online ELISE tutorials forall new UNSW students: http://subjectguides.library.unsw.edu.au/elise