350 likes | 423 Views
Monday, October 15 Statistical Inference and Probability. “I am not a crook.”. Population. Sample. You take a sample. High Stakes Coin Flip. High Stakes Coin Flip. Could your professor be a crook?. High Stakes Coin Flip. Could your professor be a crook?. Let’s do an experiment.
E N D
Monday, October 15 Statistical Inference and Probability “I am not a crook.”
Population Sample You take a sample.
High Stakes Coin Flip Could your professor be a crook?
High Stakes Coin Flip Could your professor be a crook? Let’s do an experiment.
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.
Some Steps in Hypothesis Testing Step 1. Assume that the professor is fair, i.e., that P(Win) = .5
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.
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
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.
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.
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)
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.
= 1/6 = .17 Rolling a six (6) Six possible values (1,2,3,4,5,6)
What’s the probability of rolling a dice and getting an even number?
= 3/6 = .50 Rolling an even (2, 4, 6) Six possible values (1,2,3,4,5,6)
What the probability that your first (or next) child will be a girl?
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
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
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
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?
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
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
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
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
GAUSS, Carl Friedrich 1777-1855 http://www.york.ac.uk/depts/maths/histstat/people/
1 f(X) = Where = 3.1416 and e = 2.7183 e-(X - ) / 2 2 2 2
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.
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?