7. (10pt) Gmail categorizes incoming emails into 3 groups, “Primary”, “Social” and “Promotion”. Assume 70% of my emails are primary, 20% are social and 10% are promotion. Meanwhile, I searched the keyword ”sales” in 3 groups and found out that 5% primary emails, 30% social emails and 95% promotion emails include this keyword respectively. Suppose an incoming email contains the keyword “sales”, what is the probability of being categorized as “primary”? 8. (10pt) Target is a big US retailer. By analyzing female consumer’s purchase history, data scientists at Target could predict whether she is pregnant or not. Following is a true story cited from New York Time.1 A man walked into a Target outside Minneapolis and demanded to see the manager. He was clutching coupons that had been sent to his daughter, and he was angry, according to an employee who participated in the conversation. “My daughter got this in the mail!” he said. “She’s still in high school, and you’re sending her coupons for baby clothes and cribs? Are you trying to encourage her to get pregnant?” The manager didn’t have any idea what the man was talking about. He looked at the mailer. Sure enough, it was addressed to the man’s daughter and contained advertisements for maternity clothing, nursery furniture and pictures of smiling infants. The manager apologized and then called a few days later to apologize again. 1 http://www.nytimes.com/2012/02/19/magazine/shopping-habits.html?_r=2&hp=&pagewanted=all 2 On the phone, though, the father was somewhat abashed. “I had a talk with my daughter,” he said. “It turns out there’s been some activities in my house I haven’t been completely aware of. She’s due in August. I owe you an apology.” In 2012, there were 3,988,076 new births in the US and US female population was 125.9 million. Assume all pregnancy leads to birth and all female population is in reproductive age. Please use this information to approximately calculate the prior, the probability of pregnancy in female population. Target identified that purchasing 25 specific products together can indicate pregnancy. Suppose data scientists at Target found that among pregnant female customers, 95% of them purchase these 25 products all together and among non-pregnant female customers, 0.5% of them purchase these 25 products together. Using Bayesian inference to calculate the probability that a woman is pregnant if Target observes that she purchases all these 25 products. 9. (10pt) To track how users are playing a newly launched game app, we random select 15 players and record how many minutes they spend on this game in one day, which are listed as below. 8.6; 9.4; 7.9; 6.8; 8.3; 7.3; 9.2; 9.6; 8.7; 11.4; 10.3; 5.4; 8.1; 5.5; 6.9 Please construct a 95% confidence interval of average time consumers spend with this game daily.
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