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

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: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.

“Reality”

H0 True H0 FalseD

ecis

ion Reject H0

Don’t Reject H0

“Reality”

H0 True H0 FalseD

ecis

ion Reject H0

Don’t Reject H0 Yeah!

Yeah!

Yeah!

Type I Error

“Reality”

H0 True H0 FalseD

ecis

ion Reject H0

Don’t Reject H0 Yeah!

Yeah!

Yeah!

Type I Error

Yeah!

Type II Error

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?

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?

Number of HeadsProbability

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 HeadsProbability

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?

Unimodal

Number of HeadsProbability

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?

Symmetrical

Number of HeadsProbability

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?

Two tails

GAUSS, Carl Friedrich 1777-1855

http://www.york.ac.uk/depts/maths/histstat/people/

f(X) =

Where = 3.1416 and e = 2.7183

1

2

e-(X - ) / 2 2 2

Normal Distribution

UnimodalSymmetrical34.13% 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?