Review SessionChapter 2-5
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
Chapter 3• Probability and Random variable• Continuous random variable• Discrete random variable• Multiple random variables
Probability• Some dentitions : Experiment, Sample space, Event• Fundamentals of set theory : Union, Intersection, Complement,
Mutually Exclusive• Three Conditions of probability
Properties of probability
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 0.45. The probability that you must stop at at least one of the signals is 0.50. What is the probability that you must stop at both signals?Define events A =(Stop at first signal), B =(Stop at second signal)
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 0.45. The probability that you must stop at at least one of the signals is 0.50. 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)
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 0.45. The probability that you must stop at at least one of the signals is 0.50. What is the probability that you must stop exactly one signal?Define events A =(Stop at first signal), B =(Stop at second signal)
Conditional Probability and Independence
Example: A red die and a white die are rolled. Define the eventsA = 4 on red dieB = Sum of two dice is odd
Are these two events independent?P(A) =
a) 1/6b) 1/4
Example: A red die and a white die are rolled. Define the eventsA = 4 on red dieB = Sum of two dice is odd
Are these two events independent?P(B) =
a) 1/2b) 1/4
Example: A red die and a white die are rolled. Define the eventsA = 4 on red dieB = 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 ) =
a) 1/12b) 1/14
Example: A red die and a white die are rolled. Define the eventsA = 4 on red dieB = Sum of two dice is odd
Are these two events independent?a) nob) yes
Example:Toss a coin three times. Let p be the probability of obtaining a head on each toss. Find P {HHT}
Define the eventsA=Head is observed on the first tossB=Head is observed on the second tossC=Tail is observed on the third toss
Then A ∩ B ∩ C = {HHT}
Example con’tToss 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. ThusP {HHT} = P(A ∩ B ∩ C)
= P(A)P(B)P(C) =p x p x (1-p)
Random variable
Examples: random variableSuppose f(x) = for -1 < x < 1 and f(x) = 0 otherwise. Determine C and find the following probabilities.
Examples: random variable
Random variable
Examples: random variableLet X denote the number of patients who suffer an infection within a floor of a hospital per month with the following probabilities:
Examples: random variable• Verify that the function f(x) is a probability mass function, and
determine the requested values.
Continuous Distribution
Examples: random variableReview homework 2 and 4
Discrete Distribution
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. Theprobability 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?
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?
Example: Poisson process (HW 5)• Poisson distribution• Poisson process• Normal approximation of Poisson distribution
Linear Combination of R.V.s (HW 6)
Central limit theorem (HW 6)
Chapter 4• Point estimation• Hypothesis test for one population• Confidence interval for one population• Goodness of fit test
Point estimation
Hypothesis Testing
One-sample Z test
Sample size
One-sample T test
One-sample chi-square test
One-sample approximated Z test
Testing for goodness of fit
Chapter 5• Hypothesis test for two populations• Confidence interval for two populations
Confidence Interval for
*sample statistic z SE
From original data
Z: from N(0,1) or
T: from T-dist
Large sample size or from Normal populationCaution: the multiplier depends on the significance level
Formula for p-values
From H0
sample statistic null valueSE
z
From original data
Compare z to N(0,1) or t to T distribution for p-value
Large sample size or from Normal population
Caution: The direction of the tail depends on alternative hypothesis
Decision making for two samplesParameter Distribution Standard Error (CI) Standard Error (Test)
Difference in Proportions Normal
Difference in MeansVariance Known Normal
Difference in MeansVariance Unknown but same
Pooled t, df = -2
Difference in MeansVariance Unknown but diff.
Unpooled t, df = min(n1, n2) – 1
Difference in Means (Paired) t, df = n – 1