Introduction to
Probability Theory
Michael Daniel Lucagbo
Stat 101
Objectives
n
Learn the basic terms and concepts in
probability theory.
n
n
Explain the methods of assigning probabilities
and know the properties of the probability
function.
Compute for c
Formula Sheet
Preliminaries:
2
=1( )2
=1 2 (=1 )
=
=
1
( 1)
Estimation: Single Population
Common Point Estimators under Simple Random Sampling Without Replacement
Parameter
Point Estimator
Mean:
Sample Mean:
=1
=
Standard Error of the sample mean:
Esti
Probability2
TheNormalDistribution
TheNormalDistribution
A continuous random variable X is said to be normally
distributed if its density function is given by:
TheNormalDistribution
The graph of the normal distribution is called the normal
curve
Propertie
Measurement is the process of determining
the value or label of the variable based on
what has been observed.
The interpretation of the values in the data will depend on the
measurement system used.
Properties:
a.
The numbers in the system are used to
cl
EXERCISE ON GROUPED DATA APPROXIMATIONS
Answers and Solutions
Time-pressured Exercise: Good for 30 minutes
Maximum Score: 35 pts.
GIVEN:
FREQUENCY DISTRIBUTION OF SCORES OF A SAMPLE OF STAT 131 STUDENTS
FINAL GRADE in %
31-40
41-50
51-60
61-70
71-80
81-90
TheBinomialDistribution
BinomialDistribution
A binomial experiment is one that possesses the
following properties:
The experiment consists of n identical trials
Each trial results in one of two outcomes, a success or a
failure
The probability of success o
MeasurementandDataCollection
Methods
MichaelDanielLucagbo
Measurement
How do we measure a persons age?
How can we measure poverty?
How can we measure educational attainment?
Objectives
Be familiar with the four levels of measurement.
Compare, contrast and
MeasuresofLocation
Objectives
Know how to compute for measures of location using
the weighted average estimate.
Know how to interpret the measures of location.
MeasuresofLocation
Measures of location are values below which a
specified fraction of the obse
SamplingandSamplingTechniques
Stat101
MichaelDanielLucagbo
Are there times when it is better to take a sample
rather than study the entire population?
Is it possible to make good conclusions about the
population, even if we just take a small sample of it
Boxplot and Stem-andLeaf Display
Objectives
n
Know how to construct a boxplot and a
stem-and-leaf display.
n
Identify features of the data set that are
visible from the boxplot and stem-and-leaf
display.
Boxplot
n
The box-and-whisker plot or boxplot is a
Inferential Statistics
Objectives
n
n
n
n
n
Learn the terms and concepts in hypothesis testing.
Know the steps in conducting a hypothesis test.
Formulate real-life problems in terms of hypothesis
testing problems.
Perform hypothesis tests for the mean and
Inferential Statistics
n
n
n
n
n
Know the assumptions of the chi-square test for
independence.
Perform a chi-square test for independence.
Learn the concept of correlation.
Compute for and interpret the coefficient of
correlation.
Perform the test of sign
Simple Linear Regression
Equation of a Straight Line
y = o + 1x
where 0 = y-intercept ; the value of y when x=0
slope of the line; change in y for a 1-unit increase in x
Deterministic Model vs Probabilistic Model
The linear model y = o + 1x is said to be
Chi-square Test and
Correlation
Statistics 101 - Villejo
Objectives
Know the assumptions of the chi-square test for
independence.
Perform a chi-square test for independence.
Learn the concept of correlation.
Compute for and interpret the coefficient o
CHAPTER 2
Collection and Presentation of Data
2.1. Preliminaries
Variable - characteristic or attribute of persons or objects which can assume different values or labels
for different persons or objects under consideration.
Observation - realized value of
Stat 101 SJV
Instructions: You will submit your answers for this homework on August 26. Write your answers on a yellow pad.
1. For each of the following research situations, identify the level of measurement of all variables and indicate whether
they are
Measures of
Location
Lecture Outline
Introduction
Percentiles
Quartiles
Deciles
Objectives
Know how to compute for and interpret the
measures of location.
Measures of Location
A measure of location provides
information on the percentage of
observati
Measures of
Dispersion
Lecture Outline
Introduction
Measure of Absolute Dispersion
Measure of Relative Dispersion
Range
Variance and Standard Deviation
Z-score
Coefficient of Variation
Motivation
Consider the following measurements, in liters, for
Measures of Central
Tendency
Lecture Outline
Introduction
Mean
Median
Mode
Comparison of the three
measures
Objectives
Know how to compute for and interpret the
measures of central tendency.
Identify the situations when a measure of central
tendenc
Stat 101 - SJVVillejo
1st Exam Reviewer
1. It is claimed that an automobile is driven on the average less than 20, 000 kilometers per year.
To test this claim, a random sample of 100 owners of that automobile kept a record of the
kilometers they traveled
PRACTICE PROBLEMS
Classify the following statements as belonging to
the area of descriptive statistics or inferential
statistics
As a result of recent cutbacks by the oil-producing
nations, we can expect the price of gasoline to double
in the next year.
STATISTICS
How to make sense of all these data? People
should be worried about how we train the
next generation, not just of scientists, but
people in the government and industry
- Alex Szalay, astrophysicist
We are at a different period because of so
muc
Chapter 5
Raw data - data in their original form
Array an ordered arrangement of data
according to magnitude. Also called sorted
data or ordered data.
Both raw data and array are called as ungrouped
data.
RAW DATA
ARRAY
2.75
1.25
1.75
2.75
1.50
1.75
2.25
Chapter 4
Textual Presentation incorporates
important figures in a paragraph of text.
It only includes the most important summary
statistics with additional explanation of relevance.
(Source: NSO)
The unemployment rate in April 2009 was
estimated at 7.5
Measures of
Skewness
Measures of Skewness
A measure of skewness is a single
value that indicates the degree and
direction of assymetry. It is a summary
measure describing the shape of the
distribution of data or the shape of the
frequency histogram from
Boxplot and Stemand-Leaf Display
Objectives
Know how to construct a boxplot and a
stem-and-leaf display.
Identify features of the data set that are
visible from the boxplot and stem-and-leaf
display.
Boxplot
The box-and-whisker plot or boxplot is a
gra
Reliable Estimator a statistic whose value
do not vary much from one sample to
another
Unbiased Estimator a statistic whose value
from one sample to another sample on the
average is equal to the parameter
Sampling
is more economical.
Requires less time
Inferential Statistics
Objectives
See the connection between probability
theory and inferential statistics.
n Know how to compute for and interpret
point and interval estimates for one
population.
n Know how to compute for the sample size
under certain as
Chapter 4:
Measures of Dispersion and Skewness
Dispersion
Why measure dispersion?
It is still possible for two or more data sets that have the same center to differ in other aspects.
Measures of dispersion
indicate the extent to which individual items i
CHAPTER 3
Measures of Central Tendency
Summary Measures
A single value that we compute from a collection of measurements in order to describe one of
the collections particular characteristics.
Measures of central tendency:
Provides information about cente
Chapter 6
Random Variables and Probability Distributions
Random Variable
A function whose value is a real number that is determined by each sample point in the sample
space.
An uppercase letter, say X, will be used to denote a random variable and its co
Chapter 5
Probability
Abstract Model
Description of the essential properties of a phenomenon that is formulated in mathematical terms.
Deterministic model describes the phenomenon through known relationships among the
states and events, in which a given
Chapter 8
Estimation
POINT ESTIMATION
Point estimator single statistic whose realized value is used to estimate the population
parameter.
Point estimate A realized value of an estimator.
Optimal Properties of a Point Estimator
1. Unbiased Estimator (Expec