Monday, October 15 Statistical Inference and Probability

1. Monday, October 15 Statistical Inference and Probability “I am not a crook.”

2. Population Sample You take a sample.

3. High Stakes Coin Flip

4. High Stakes Coin Flip Could your professor be a crook?

5. High Stakes Coin Flip Could your professor be a crook? Let’s do an experiment.

6. The Coin Flip Experiment • Question: Could the professor be a crook? • Let’s do an experiment. • Make assumptions about the professor. • Determine sampling frame. • Set up hypotheses based on assumptions. • Collect data. • Analyze data. • Make decision whether he is or is not a crook.

7. Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) = .5

8. Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) = .5 Step 2. Set up hypotheses: H0: He is not a crook. H1: He is a crook.

9. Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) = .5 Step 2. Set up hypotheses: H0: He is not a crook. H1: He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at .05

10. Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) = .5 Step 2. Set up hypotheses: H0: He is not a crook. H1: He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at .05 Step 4. Decide on a sample, e.g., 6 flips.

11. Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) = .5 Step 2. Set up hypotheses: H0: He is not a crook. H1: He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at .05 Step 4. Decide on a sample, e.g., 6 flips. Step 5. Gather data.

12. Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) = .5 Step 2. Set up hypotheses: H0: He is not a crook. H1: He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at .05 Step 4. Decide on a sample, e.g., 6 flips. Step 5. Gather data. Step 6. Decide whether the data is more or less probable than  . E.g., the probability of 6 consecutive wins based on the assumption in Step 1 is .016. (.5 x .5 x .5 x .5 x .5 x .5 = .016)

13. Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) = .5 Step 2. Set up hypotheses: H0: He is not a crook. H1: He is a crook. Step 3. Determine the risk that you are willing to take in making an error of false slander,  (alpha), often at .05 Step 4. Decide on a sample, e.g., 6 flips. Step 5. Gather data. Step 6. Decide whether the data is more or less probable than  . E.g., the probability of 6 consecutive wins based on the assumption in Step 1 is .016. (.5 x .5 x .5 x .5 x .5 x .5 = .016) Step 7. Based on this evidence, determine if the assumption that Hakuta is fair should be rejected or not.

17. What’s the probability of rolling a dice and getting 6?

18. = 1/6 = .17 Rolling a six (6) Six possible values (1,2,3,4,5,6)

19. What’s the probability of rolling a dice and getting an even number?

20. = 3/6 = .50 Rolling an even (2, 4, 6) Six possible values (1,2,3,4,5,6)

21. What the probability that your first (or next) child will be a girl?

22. What is the probability of flipping 8 heads in a row?

23. What is the probability of flipping 8 heads in a row? .5 x .5 x .5 x .5 x .5 x .5 x .5 x .5 or .58 = .004

24. What is the probability of flipping 8 heads in a row? .5 x .5 x .5 x .5 x .5 x .5 x .5 x .5 or .58 = .004 Formalized as: The probability that A, which has probability P(A), will occur r times in r independent trials is: P(A)r

25. So, you decide to conduct a case study of 3 teachers, sampling randomly from a school district where 85% of the teacher are women. You end up with 3 male teachers. What do you conclude? P(males) three times = P(males)3 = .153 = .003

26. So, you decide to conduct a case study of 3 teachers, sampling randomly from a school district where 85% of the teacher are women. You end up with 3 male teachers. What do you conclude? P(males) three times = P(males)3 = .153 = .003 If you had ended up with 3 female teachers, would you have been surprised?

27. What do you notice about this distribution? Number of Heads Probability 0 1/64 =.016 1 6/64 =.094 2 15/64 =.234 3 20/64 =.312 4 15/64 =.234 5 6/64 =.094 6 1/64 =.016 ___________ 64/64 =1.00

28. What do you notice about this distribution? Number of Heads Probability 0 1/64 =.016 1 6/64 =.094 2 15/64 =.234 3 20/64 =.312 4 15/64 =.234 5 6/64 =.094 6 1/64 =.016 ___________ 64/64 =1.00 Unimodal

29. What do you notice about this distribution? Number of Heads Probability 0 1/64 =.016 1 6/64 =.094 2 15/64 =.234 3 20/64 =.312 4 15/64 =.234 5 6/64 =.094 6 1/64 =.016 ___________ 64/64 =1.00 Symmetrical

30. What do you notice about this distribution? Number of Heads Probability 0 1/64 =.016 1 6/64 =.094 2 15/64 =.234 3 20/64 =.312 4 15/64 =.234 5 6/64 =.094 6 1/64 =.016 ___________ 64/64 =1.00 Two tails

31. GAUSS, Carl Friedrich 1777-1855 http://www.york.ac.uk/depts/maths/histstat/people/

32. 1 f(X) = Where = 3.1416 and e = 2.7183 e-(X - ) / 2  2 2  2

33. Normal Distribution Unimodal Symmetrical 34.13% of area under curve is between µ and +1  34.13% of area under curve is between µ and -1  68.26% of area under curve is within 1  of µ. 95.44% of area under curve is within 2  of µ.

35. Some Problems • If z = 1, what % of the normal curve lies above it? Below it? • If z = -1.7, what % of the normal curve lies below it? • What % of the curve lies between z = -.75 and z = .75? • What is the z-score such that only 5% of the curve lies above it? • In the SAT with µ=500 and =100, what % of the population • do you expect to score above 600? Above 750?