1. Review Session
Chapter 2-5

2. Chapter 2 Population and Sample
Population: The entire collection of all objects under study
Sample: Any subset of the population
Data summary statistics and data display
Location : Sample mean, Median, Quartiles
Spread: Range, Inter-quantile range (IQR), Sample Variance, Sample standard deviation
Data display: Dot-diagram, Stem-and-leaf diagram, Histogram, Box-plot
Scatter diagram and sample correlation coefficient
Scatter diagram is graphical description for looking at relationship between two variables
Sample correlation coefficient numerical summary for linear relationship between two variables

3. Chapter 3 Probability and Random variable Continuous random variable
Discrete random variable
Multiple random variables

4. Probability Some dentitions : Experiment, Sample space, Event
Fundamentals of set theory : Union, Intersection, Complement, Mutually Exclusive
Three Conditions of probability

5. Properties of probability

6. Example: probability
When driving to campus, there are two intersections with traffic lights on your way. The probability that you must stop at the first signal is 0.30 and the probability that you must stop at the second signal is The probability that you must stop at at least one of the signals is What is the probability that you must stop at both signals? Define events A =(Stop at first signal), B =(Stop at second signal)

7. Example: probability
When driving to campus, there are two intersections with traffic lights on your way. The probability that you must stop at the first signal is 0.30 and the probability that you must stop at the second signal is The probability that you must stop at at least one of the signals is What is the probability that you must stop at the first signal but not at the second one? Define events A =(Stop at first signal), B =(Stop at second signal)

8. Example: probability
When driving to campus, there are two intersections with traffic lights on your way. The probability that you must stop at the first signal is 0.30 and the probability that you must stop at the second signal is The probability that you must stop at at least one of the signals is What is the probability that you must stop exactly one signal? Define events A =(Stop at first signal), B =(Stop at second signal)

9. Conditional Probability and Independence

10. Example: A red die and a white die are rolled. Define the events
A = 4 on red die
B = Sum of two dice is odd
Are these two events independent?
P(A) =
1/6
1/4

11. Example: A red die and a white die are rolled. Define the events
A = 4 on red die
B = Sum of two dice is odd
Are these two events independent?
P(B) =
1/2
1/4

12. Example: A red die and a white die are rolled. Define the events
A = 4 on red die
B = Sum of two dice is odd
Are these two events independent?
P(A∩B) = P(4 on red die and sum of two dice is odd ) =
1/12
1/14

13. Example: A red die and a white die are rolled. Define the events
A = 4 on red die
B = Sum of two dice is odd
Are these two events independent? no
yes

14. Example:
Toss a coin three times. Let p be the probability of obtaining a head on each toss. Find P {HHT}
Define the events
A=Head is observed on the first toss
B=Head is observed on the second toss
C=Tail is observed on the third toss
Then A ∩ B ∩ C = {HHT}

15. Example con’t
Toss a coin three times. Let p be the probability of obtaining a head on each toss. Find P {HHT} From the experiment, events A, B and C are independent. Thus P {HHT} = P(A ∩ B ∩ C) = P(A)P(B)P(C) =p x p x (1-p)

16. Random variable

17. Examples: random variable
Suppose f(x) = 𝐶 𝑥 2 for -1 < x < 1 and f(x) = 0 otherwise. Determine C and find the following probabilities.

18. Examples: random variable

19. Random variable

20. Examples: random variable
Let X denote the number of patients who suffer an infection within a floor of a hospital per month with the following probabilities:

21. Examples: random variable
Verify that the function f(x) is a probability mass function, and determine the requested values.

22. Continuous Distribution

23. Examples: random variable
Review homework 2 and 4

24. Discrete Distribution

25. Example:
Because not all airline passengers show up for their reserved seat, an airline sells 135 tickets for a flight that holds only 130 passengers. The
probability that a passenger does not show up is 0.08, and the passengers behave independently.
What is the probability that every passenger who shows up can take the flight?
What is the probability that the flight departs with empty seats?

26. Example:
In 1898 L. J. Bortkiewicz published a book entitled The Law of Small Numbers. He used data collected over 20 years to show that the number of soldiers killed by horse kicks each year in each corps in the Prussian cavalry followed a Poisson distribution with a mean of 0.61.
What is the probability of more than one death in a corps in a year?
What is the probability of no deaths in a corps over five years?

27. Example: Poisson process (HW 5)
Poisson distribution
Poisson process
Normal approximation of Poisson distribution

28. Linear Combination of R.V.s (HW 6)

29. Central limit theorem (HW 6)

30. Chapter 4 Point estimation Hypothesis test for one population
Confidence interval for one population
Goodness of fit test

31. Point estimation

32. Hypothesis Testing

33. One-sample Z test

34. Sample size

35. One-sample T test

36. One-sample chi-square test

37. One-sample approximated Z test

38. Testing for goodness of fit

39. Chapter 5 Hypothesis test for two populations
Confidence interval for two populations

40. Confidence Interval for 𝑯 𝟎 ≔𝒗𝒔 𝑯 𝟏 :≠
Large sample size or from Normal population
Z: from N(0,1) or
T: from T-dist
Caution:
the multiplier depends on the significance level
From original data

41. Large sample size or from Normal population
Formula for p-values
Large sample size or from Normal population
From original data
From H0
Caution:
The direction of the tail depends on alternative hypothesis
Compare z to N(0,1) or
t to T distribution for p-value

42. Decision making for two samples
Parameter
Distribution
Standard Error (CI)
Standard Error (Test)
Difference in Proportions
Normal
Difference in Means
Variance Known
Variance Unknown but same
Pooled
t, df = 𝑛 1 + 𝑛 2 -2
Variance Unknown but diff.
Unpooled
t, df = min(n1, n2) – 1
Difference in Means (Paired)
t, df = n – 1