Wednesday, October 19

We need to start a new field! The Stanford Journal Of Body Part Estimating Ability Inaugural Special Issue: Hands. Next: Toes. Volume 1, No. 1. October, 2011

We now move from the world of description and prediction to the world of hypothesis testing and decision-making using probability…

We now turn to the world of urns…

Population Sample You take a sample, and use a probability model to make a decision about its noteworthiness.

Wednesday, October 19 Statistical Inference and Probability “I am not a crook.”

High Stakes Coin Flip

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: H 0 : He is not a crook. H 1 : 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: H 0 : He is not a crook. H 1 : 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: H 0 : He is not a crook. H 1 : 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: H 0 : He is not a crook. H 1 : 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: H 0 : He is not a crook. H 1 : 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: H 0 : He is not a crook. H 1 : 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 Kenji is fair should be rejected or not.

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

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

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

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

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.5 8 =.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.5 8 =.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 =.15 3 =.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 =.15 3 =.003 If you had ended up with 3 female teachers, would you have been surprised?

Number of HeadsProbability 0 1/64= /64= /64= /64= /64= /64= /64=.016 ___________ 64/64=1.00 What do you notice about this distribution?

Number of HeadsProbability 0 1/64= /64= /64= /64= /64= /64= /64=.016 ___________ 64/64=1.00 What do you notice about this distribution? Unimodal

Number of HeadsProbability 0 1/64= /64= /64= /64= /64= /64= /64=.016 ___________ 64/64=1.00 What do you notice about this distribution? Symmetrical

Number of HeadsProbability 0 1/64= /64= /64= /64= /64= /64= /64=.016 ___________ 64/64=1.00 What do you notice about this distribution? Two tails

GAUSS, Carl Friedrich

f(X) = Where  = and e =  2  e -(X -  ) / 2  22

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 % 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?