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Transcript
Quantitative Business Methods for
Decision Making
Estimation and
Testing of
Hypotheses
Lecture Outlines
Estimation
Confidence interval for estimating
means
 Confidence interval for predicting a new
observation
 Confidence interval for estimating
proportions

403.6
2
Lecture Outlines (con’t)
Hypothesis Testing
Null and alternative hypotheses
 Decision rules (Tests) and their level of
significance
 Type I and Type II errors
 Tests of hypotheses for comparing means
 Tests of hypotheses for comparing
proportions
403.6
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
Estimating a Population Mean 
Population mean  is estimated by x ,
the sample mean
s
 Standard error of x , i.e. s x 

n
will decrease as n gets large.
403.6
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Confidence Interval for
 

Estimating  if   n is known
x
 is
With a 95% degree of
confidence


estimated within ( X  2
X  2 )
x
x
written as X  2 x Or more accurately
by X  1.96
x
403.6
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Confidence Interval for
 

 
 is not known
n


if
x
Use x  ts x instead of x  1.96 x ,
remember s x 
s
n
, and
“t” is 95th% percentile of the t
distribution with (n-1) degrees of
freedom.
403.6
6
An Illustration
Suppose n= 26. Then degrees of freedom
(d.f.) = n-1 = 25.
A two-sided 95% degree of C.I. is computed
by x  2.064s x
But, for a one-sided 95% C.I. , t = 1.711 instead
of 2.064
403.6
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Assumptions and Sample Size for
Estimation of the mean
The population should be normally (at least
close to) distributed. If skew, then median is
an appropriate measure of the center than the
mean.
To estimate mean with a specified margin of
error (m.e.), take a random sample of size n
z 2 2
n
2
m.e. 403.6
8
Prediction Interval for a New
Observation on X
Prediction Interval for a new observation is given by
1
x  ts 1 
n
403.6
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Confidence Interval for a
Population Proportion 
Let  denote the proportion of items in a
population having a certain property
An estimate of  is the binomial
X
proportion: p 
, What is X?
n
For a C.I. for

, use p  ts p
403.6
10
Confidence Intervals for the
Proportion (con’t)
For estimating  ,“t” is the percentile of the
t-distribution with df (equivalently,
percentile of the standard normal
distribution), and s.e. of p is
sp 
p(1  p)
n
403.6
11
Hypotheses Testing
The hypothesis testing is a methodology
for proving or disproving researcher’s prior
beliefs.
 Statements that express prior beliefs are
framed as alternative hypotheses.
 Complementary statement to an
alternative hypothesis is called null
hypothesis.

403.6
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Null and Alternative
Ha: Researcher’s belief that are to be tested
(alternate hypothesis)
H0: Complement of Ha (Null hypothesis)
403.6
13
Statistical Decision
A decision will be either:
Reject H0 (Ha is proved)
or
Do not reject H0 (Ha is not
proved)
403.6
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Hypothesis Testing Methodology for
the mean 
Depending upon what an investigator
believes a priori, an alternative hypothesis
is formulated to be one of the followings:
Ha :   0
1.
2.
Ha :   0
3.
Ha :   0
one-sided
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A Test Statistic
Regardless of what an alternative hypothesis
about the mean is formulated, the decision
rule is defined by a t- statistic:
X  0
t
Sx
403.6
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Decision Rules for Testing
Hypotheses About the Mean
Hypotheses
1.
Ho:  = o
Ha:   o
2.
Ho:   o
Ha:   o
3.
Ho:   o
Ha:   o
Decision Rule
At  level of significance, reject Ho in favor of Ha if
either t-statistic t 2 or  -t 2
At  level of significance, reject Ho in favor of Ha if
t-statistic t

At  level of significance, reject Ho in favor of Ha if
t-statistic t 403.6
17
Type I and Type II Errors
Decision taken is
Accept H0
Accept H1
----------------------------------------------------------------------Suppose correct
Type I Error
H0 is true
decision
(wrong decision)
-----------------------------------------------------------------------Suppose
Type II Error
correct
H1 is true
(wrong decision)
decision
Type I error : reject H0 if H0 is true.
Type II error : Do not reject H0 when H0 is false.
403.6
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Comparing Two Means
X population
X ,
X
Y population
Y ,
Y
Consider
1. H0: X - Y = 
2. H0: X - Y  
Ha: X - Y  ,
Ha: X - Y  ,
3. H0: X - Y  
Ha: X - Y  .
The reference number  is a specified amount for comparing the
difference between two means. There are two distinct practical
situations resulting in samples on X and Y.
403.6
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Two Sampling Designs
•Paired Sample
•Two independent Samples
403.6
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Paired Sample




Two variables X and Y are observed for each unit
in the sample to measure the same aspect but
under two different conditions.
Thus, for n randomly selected units, a sample of
n pairs (X, Y) is observed.
Compute d differences: X1-Y1= d1, X2-Y2= d2, etc.
d
and then mean
Compute Sd of differences
403.6
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Paired Samples (con’t)
Compute
Sd 
sd
n
d 
t-statistic: t - statistic 
Sd
403.6
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Paired Samples (con’t)
•Reject H0 if absolute value of t-statistic is more than
the desired percentile of the t-distribution.
•Alternatively, find the p value of the t-statistics and
reject H0 if the p value is less than the desired
significance level.
403.6
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Two Independent
(Unpaired) Samples
Populations of variables X and Y (for
example, males salary X and females
salary Y).
 Take samples independently on X and Y.
 Compute X, S , Y, S
x
y
 Compute pooled standard deviation

S
n1  1S2x  n2  1S2y
n1  n2  2
403.6
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Unpaired Samples (con’t)

Compute


SE of X  Y  S x  y

Finally, compute
t-statistic=
1
1
S

n1 n 2
X  Y   
Sx y
Use p value to reach a decision
403.6
25
Comparing Proportions  1 and  2
To estimate  1   2  in a 95% C.I.,
compute,
 p1 1  p1  p2 1  p2  

 p1  p2   1.96 

n2
 n1

403.6
26
Comparing Two Proportions

For testing hypothesis about the
difference  1   2  , compute
n1 p1  n2 p2
n  n1  n2 , p 
n
and
t-statistic=
 p1 
p2 
p (1  p )
n
403.6
27