Question
Establish the following recursion relations for means and variances.
Let X n
2
and Sn be the mean and variance, respectively,
. ,XnX. Then suppose
1, . .of
another observation
X becomes available.
Prove the following:
n+1
X n+1 + nXn
(a) Xn+1 =
n
HBS Quantitative Methods
Summaries
Summaries
HBS Quantitative Methods
Table of Contents
Working with Data Histograms . 1
Central Values for Data . 1
Variability Standard Deviation . 2
The Coefficient of Variation . 2
Relationship between two variables. 3
CoefficientofVariation
Thecoefficientofvariationindicateshowlargethe
standarddeviationisinrelationtothemean.
Thecoefficientofvariationiscomputedasfollows:
s
100 %
x
TanweerulIslam
100 %
fora
sample
fora
population
Slide
1
Variance,StandardDeviation,
An
Probability (Class Activity)
1. Given:
S=cfw_E1,E2,E3,E4,E5,E6,E7,E8,E9,E10
A= cfw_E1, E3, E7, E9
B= cfw_E2, E3, E8, E9
Find
(a). A complement and B complement
(b).
(c).
(d). Is union of A and B is collectively exhaustive?
(e). Are these events mutually e
Normal Distribution Practice Exercises
Instructions:
1. Answer each exercise, using the Standard Normal Table as appropriate.
2. You are encouraged to work together.
3. For each exercise, a properly-labeled sketch is recommended.
Exercises
Part 1: Using t
Measures of Shape
Skewness
Generally, skewness may be indicated by looking
at the sample histogram or by comparing the mean
and median.
This visual indicator is imprecise and does not take
into consideration sample size n.
TanweerulIslam
Slide
1
Measures
HypothesisTesting
1
Hypothesis Testing
What is a Hypothesis?
A statement about population that may or may not be true.
Examples:
O The average nicotine content of a new brand of cigarette is 0.05
milligrams.
O The average life of a new battery is over 72
Hypothesis Testing
Question 1:
A random sample of 541 consumers was asked to respond on a scale from one (strongly
disagree) to five (strongly agree) to the assertion that a limit should be placed on the amount of
punitive damages awarded for product liab
Descriptive Statistics: Numerical Measures
I
I
I
Measures of Location
Measures of Variability
Measure of Relative Position
Tanweer-ulTanweer-ul-Islam
Slide 1
Measures of Location
I
Mean
I
Median
Mode
I
If the measures are computed
for data from a sample,
Answers of Probability Questions
Q.1
Probability of each outcome =1/6
Probability of even outcome= 3/6=1/2
Probability of odd outcome = 3/6 =1/2
Q.2
P (A) =Probability of 2= 1/6
P (B) = Probability of 5 = 1/6
P(AUB)= 1/6+1/6-0= 2/6=1/3
Q.3
a. 6/23
b. 5/23
CoefficientofVariation
Thecoefficientofvariationindicateshowlargethe
standarddeviationisinrelationtothemean.
Thecoefficientofvariationiscomputedasfollows:
s
100 %
x
TanweerulIslam
100 %
fora
sample
fora
population
Slide
1
Variance,StandardDeviation,
An
Basic Probability
Concepts & Applications
Chap 2-1
Important Terms
Random Experiment a process leading to an
uncertain outcome
A coin is thrown
A consumer is asked which of two products he
or she prefers
The daily change in an index of stock market
pri
One-Way ANOVA
One-Way
OneWayAnalysisofVariance
1
One-Way ANOVA
One-Way
The one-way analysis of variance is used
The
to test the claim that three or more
population means are equal
population
This is an extension of the two
This
independent samples t-test
Normal Probability Distribution
Normal
The
normal probability distribution is the most
important distribution for describing a continuous
random variable.
It
is widely used in statistical inference.
Tanweer-ul-Islam
1
NormalProbabilityDistribution
s
Ithas
Measures of Shape
Skewness
Skewness
Generally, skewness may be indicated by looking
at the sample histogram or by comparing the mean
and median.
This visual indicator is imprecise and does not take
into consideration sample size n.
TanweerulIslam
Slide
HypothesisTesting
1
Hypothesis Testing
What is a Hypothesis?
A statement about population that may or may not be true.
Examples:
O The average nicotine content of a new brand of cigarette is 0.05
milligrams.
O The average life of a new battery is over 72
Chi-Square Test for
Independence
Characteristics of the Chi-Square Distribution
1. It is not symmetric.
Characteristics of the Chi-Square Distribution
1. It is not symmetric.
2. The shape of the chi-square distribution
depends upon the degrees of freedom,