1. Statistical Inference and Probability
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. Could your professor be a crook?
High Stakes Coin Flip
Could your professor be a crook?

5. Could your professor be a crook?
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. Rolling a six (6)
Six possible values (1,2,3,4,5,6)
= 1/6 = .17

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

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

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

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