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Statistical Inference and Hypothesis Testing Dennis Sun Data 301 Statistical Inference probability Population / Box Sample / Data statistics The goal of statistics is to infer the unknown population from the sample. Probability vs. Statistics Probability: I have a fair coin. If I toss it 100 times, how many heads will I get? 0 1 100 draws with replacement ??? Statistics: I have a coin. I do not know if it is fair or not. I toss it 100 times and get 60 heads. Is the coin fair or not? ? ... ? 100 draws with replacement 0 , 1 , 1 , ..., 0 | {z } 60 1 s Hypothesis Testing One method of statistical inference is hypothesis testing. Idea: Assume a box model. We can see whether the data that was observed is consistent with that box. Example: (continued from previous slide) We don’t know whether the coin is fair or not. But let’s assume that it is. Then the data should be like 100 draws with replacement from the box 0 1 . Here’s the distribution of the number of 1 s from that box: Hypothesis Testing One method of statistical inference is hypothesis testing. Idea: Assume a box model. We can see whether the data that was observed is consistent with that box. Example: (continued from previous slide) We don’t know whether the coin is fair or not. But let’s assume that it is. Then the data should be like 100 draws with replacement from the box 0 1 . Here’s the distribution of the number of 1 s from that box: If the coin really were fair, we would expect to see 60 heads or more just 2.8% of the time. The Logic of Hypothesis Testing We observed 60 heads in 100 tosses. Is the coin fair or not? 1 Assume that it is fair. (This is called the null hypothesis.) 2 If it is fair, then there is just a 2.8% chance that we observe 60 heads or more. (This probability is called the p-value.) We now have a choice: 3 • Believe the null hypothesis and accept that a 1-in-35 event has just occurred. • Reject the null hypothesis and conclude that something else is going on. The smaller the p-value, the more unlikely the event you have to accept to believe the null hypothesis.