Review Session Chapter 2-5

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