SlideShare a Scribd company logo
1
Chapter 10
Comparisons Involving Means
µµ11
== µµ22
??
ANOVAANOVA
Estimation of the Difference between the Means of
Two Populations: Independent Samples
Hypothesis Tests about the Difference between the
Means of Two Populations: Independent Samples
Inferences about the Difference between the Means
of Two Populations: Matched Samples
Introduction to Analysis of Variance (ANOVA)
ANOVA: Testing for the Equality of k Population
Means
2
Estimation of the Difference Between the Means
of Two Populations: Independent Samples
Point Estimator of the Difference between the Means
of Two Populations
Sampling Distribution
Interval Estimate of µ1−µ2: Large-Sample Case
Interval Estimate of µ1−µ2: Small-Sample Case
x x1 2−
3
Point Estimator of the Difference Between
the Means of Two Populations
Let µ1 equal the mean of population 1 and µ2 equal the
mean of population 2.
The difference between the two population means is
µ1 - µ2.
To estimate µ1 - µ2, we will select a simple random
sample of size n1 from population 1 and a simple
random sample of size n2 from population 2.
Let equal the mean of sample 1 and equal the
mean of sample 2.
The point estimator of the difference between the
means of the populations 1 and 2 is .
x x1 2−
x1 x2
4
E x x( )1 2 1 2− = −µ µ
n Properties of the Sampling Distribution ofProperties of the Sampling Distribution of
• Expected ValueExpected Value
Sampling Distribution ofSampling Distribution of x x1 2−
x x1 2−
5
Properties of the Sampling Distribution of
– Standard Deviation
where: σ1 = standard deviation of population 1
σ2 = standard deviation of population 2
n1= sample size from population 1
n2 = sample size from population 2
Sampling Distribution of x x1 2−
x x1 2−
σ
σ σ
x x
n n1 2
1
2
1
2
2
2
− = +
6
Interval Estimate with σ1 and σ2 Known
where:
1 - α is the confidence coefficient (level).
Interval Estimate of µ1 - µ2:
Large-Sample Case (n1 > 30 and n2 > 30)
x x z x x1 2 2 1 2
− ± −α σ/
7
n Interval Estimate withInterval Estimate with σσ11 andand σσ22 UnknownUnknown
where:where:
Interval Estimate of µ1 - µ2:
Large-Sample Case (n1 > 30 and n2 > 30)
x x z sx x1 2 2 1 2
− ± −α/
s
s
n
s
n
x x1 2
1
2
1
2
2
2
− = +
8
Example: Par, Inc.
Interval Estimate of µ1 - µ2: Large-Sample Case
Par, Inc. is a manufacturer of golf equipment and
has developed a new golf ball that has been designed
to provide “extra distance.” In a test of driving
distance using a mechanical driving device, a sample of
Par golf balls was compared with a sample of golf balls
made by Rap, Ltd., a competitor.
The sample statistics appear on the next slide.
9
Example: Par, Inc.
Interval Estimate of µ1 - µ2: Large-Sample Case
– Sample Statistics
Sample #1 Sample #2
Par, Inc. Rap, Ltd.
Sample Size n1 = 120 balls n2 = 80 balls
Mean = 235 yards = 218 yards
Standard Dev. s1 = ___yards s2 =____ yards
x1 2x
10
Point Estimate of the Difference Between Two
Population Means
µ1 = mean distance for the population of
Par, Inc. golf balls
µ2 = mean distance for the population of
Rap, Ltd. golf balls
Point estimate of µ1 - µ2 = = 235 - 218 = 17 yards.x x1 2−
Example: Par, Inc.
11
Point Estimator of the Difference Between
the Means of Two Populations
Population 1Population 1
Par, Inc. Golf BallsPar, Inc. Golf Balls
µµ11 = mean driving= mean driving
distance of Pardistance of Par
golf ballsgolf balls
Population 1Population 1
Par, Inc. Golf BallsPar, Inc. Golf Balls
µµ11 = mean driving= mean driving
distance of Pardistance of Par
golf ballsgolf balls
Population 2Population 2
Rap, Ltd. Golf BallsRap, Ltd. Golf Balls
µµ22 = mean driving= mean driving
distance of Rapdistance of Rap
golf ballsgolf balls
Population 2Population 2
Rap, Ltd. Golf BallsRap, Ltd. Golf Balls
µµ22 = mean driving= mean driving
distance of Rapdistance of Rap
golf ballsgolf balls
µµ11 –– µµ22 = difference between= difference between
the mean distancesthe mean distances
Simple random sampleSimple random sample
ofof nn11 Par golf ballsPar golf balls
xx11 = sample mean distance= sample mean distance
for sample of Par golf ballfor sample of Par golf ball
Simple random sampleSimple random sample
ofof nn11 Par golf ballsPar golf balls
xx11 = sample mean distance= sample mean distance
for sample of Par golf ballfor sample of Par golf ball
Simple random sampleSimple random sample
ofof nn22 Rap golf ballsRap golf balls
xx22 = sample mean distance= sample mean distance
for sample of Rap golf ballfor sample of Rap golf ball
Simple random sampleSimple random sample
ofof nn22 Rap golf ballsRap golf balls
xx22 = sample mean distance= sample mean distance
for sample of Rap golf ballfor sample of Rap golf ball
xx11 -- xx22 = Point Estimate of= Point Estimate of µµ11 –– µµ22
12
95% Confidence Interval Estimate of the Difference
Between Two Population Means: Large-Sample Case,
σ1 and σ2 Unknown
Substituting the sample standard deviations for
the population standard deviation:
= ___________ or 11.86 yards to 22.14 yards.
We are 95% confident that the difference between the
mean driving distances of Par, Inc. balls and Rap, Ltd.
balls lies in the interval of _______________ yards.
x x z
n n
1 2 2
1
2
1
2
2
2
2 2
17 1 96
15
120
20
80
− ± + = ± +α
σ σ
/ .
( ) ( )
Example: Par, Inc.
13
Interval Estimate of µ1 - µ2:
Small-Sample Case (n1 < 30 and/or n2 < 30)
Interval Estimate with σ 2
Known (and equal)
where:
x x z x x1 2 2 1 2
− ± −α σ/
σ σx x
n n1 2
2
1 2
1 1
− = +( )
14
Interval Estimate with σ 2
Unknown (and assumed
equal)
where:
and the degrees of freedom for the t-distribution is
n1+n2-2.
Interval Estimate of µ1 - µ2:
Small-Sample Case (n1 < 30 and/or n2 < 30)
x x t sx x1 2 2 1 2
− ± −α /
s
n s n s
n n
2 1 1
2
2 2
2
1 2
1 1
2
=
− + −
+ −
( ) ( )
s s
n n
x x1 2
2
1 2
1 1
− = +( )
15
Example: Specific Motors
Specific Motors of Detroit has developed a new
automobile known as the M car. 12 M cars and 8 J cars
(from Japan) were road tested to compare miles-per-
gallon (mpg) performance. The sample statistics are:
Sample #1 Sample #2
M Cars J Cars
Sample Size n1 = 12 cars n2 = 8 cars
Mean = 29.8 mpg = 27.3 mpg
Standard Deviation s1 = ____ mpg s2 = ____ mpg
x2x1
16
Point Estimate of the Difference Between Two
Population Means
µ1 = mean miles-per-gallon for the population of
M cars
µ2 = mean miles-per-gallon for the population of
J cars
Point estimate of µ1 - µ2 = = ________ = ___ mpg.x x1 2−
Example: Specific Motors
17
95% Confidence Interval Estimate of the Difference
Between Two Population Means: Small-Sample Case
We will make the following assumptions:
– The miles per gallon rating must be normally
distributed for both the M car and the J car.
– The variance in the miles per gallon rating must
be the same for both the M car and the J car.
Example: Specific Motors
18
n 95% Confidence Interval Estimate of the Difference
Between Two Population Means: Small-Sample Case
Using the t distribution with n1 + n2 - 2 = ___ degrees
of freedom, the appropriate t value is t.025 = ______.
We will use a weighted average of the two sample
variances as the pooled estimator of σ 2
.
Example: Specific Motors
19
95% Confidence Interval Estimate of the Difference
Between Two Population Means: Small-Sample Case
= _____________, or .3 to 4.7 miles per gallon.
We are 95% confident that the difference between the
mean mpg ratings of the two car types is from .3 to
4.7 mpg (with the M car having the higher mpg).
s
n s n s
n n
2 1 1
2
2 2
2
1 2
2 2
1 1
2
11 2 56 7 1 81
12 8 2
5 28=
− + −
+ −
=
+
+ −
=
( ) ( ) ( . ) ( . )
.
x x t s
n n
1 2 025
2
1 2
1 1
2 5 2 101 5 28
1
12
1
8
− ± + = ± +. ( ) . . . ( )
Example: Specific Motors
20
Hypotheses
H0: µ1- µ2 < 0 H0: µ1 - µ2 > 0 H0: µ1- µ2 = 0
Ha: µ1- µ2 > 0 Ha: µ1 - µ2 < 0 Ha: µ1- µ2 ≠ 0
Test Statistic
Large-Sample Small-Sample
Hypothesis Tests About the Difference
between the Means of Two Populations:
Independent Samples
z
x x
n n
=
− − −
+
( ) ( )1 2 1 2
1
2
1 2
2
2
µ µ
σ σ
t
x x
s n n
=
− − −
+
( ) ( )
( )
1 2 1 2
2
1 21 1
µ µ
21
Hypothesis Tests About the Difference between the
Means of Two Populations: Large-Sample Case
Par, Inc. is a manufacturer of golf equipment and has
developed a new golf ball that has been designed to
provide “extra distance.” In a test of driving distance
using a mechanical driving device, a sample of Par
golf balls was compared with a sample of golf balls
made by Rap, Ltd., a competitor. The sample
statistics appear on the next slide.
Example: Par, Inc.
22
Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
– Sample Statistics
Sample #1 Sample #2
Par, Inc. Rap, Ltd.
Sample Size n1 = 120 balls n2 = 80 balls
Mean = 235 yards = 218 yards
Standard Dev. s1 = ____ yards s2 = ____ yards
Example: Par, Inc.
x1 x2
23
Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
Can we conclude, using a .01 level of
significance, that the mean driving distance of Par,
Inc. golf balls is greater than the mean driving
distance of Rap, Ltd. golf balls?
Example: Par, Inc.
24
n Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
µ1 = mean distance for the population of Par, Inc.
golf balls
µ2 = mean distance for the population of Rap, Ltd.
golf balls
•Hypotheses H0: µ1 - µ2 < 0
Ha: µ1 - µ2 > 0
Example: Par, Inc.Example: Par, Inc.
25
Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
– Rejection Rule
Reject H0 if z > ________
z
x x
n n
=
− − −
+
=
− −
+
= =
( ) ( ) ( )
( ) ( ) .
.1 2 1 2
1
2
1
2
2
2
2 2
235 218 0
15
120
20
80
17
2 62
6 49
µ µ
σ σ
Example: Par, Inc.
26
n Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
• Conclusion
Reject H0. We are at least 99% confident that the
mean driving distance of Par, Inc. golf balls is
greater than the mean driving distance of Rap,
Ltd. golf balls.
Example: Par, Inc.Example: Par, Inc.
27
Hypothesis Tests About the Difference Between the
Means of Two Populations: Small-Sample Case
Can we conclude, using a .05 level of
significance, that the miles-per-gallon (mpg)
performance of M cars is greater than the miles-per-
gallon performance of J cars?
Example: Specific Motors
28
n Hypothesis Tests About the Difference Between the
Means of Two Populations: Small-Sample Case
µ1 = mean mpg for the population of M cars
µ2 = mean mpg for the population of J cars
•Hypotheses H0: µ1 - µ2 < 0
Ha: µ1 - µ2 > 0
Example: Specific MotorsExample: Specific Motors
29
Example: Specific Motors
Hypothesis Tests About the Difference Between the
Means of Two Populations: Small-Sample Case
– Rejection Rule
Reject H0 if t > _______
(a = .05, d.f. = 18)
– Test Statistic
where:
t
x x
s n n
=
− − −
+
( ) ( )
( )
1 2 1 2
2
1 21 1
µ µ
2 2
2 1 1 2 2
1 2
( 1) ( 1)
2
n s n s
s
n n
− + −
=
+ −
30
Inference About the Difference between the
Means of Two Populations: Matched Samples
With a matched-sample design each sampled item
provides a pair of data values.
The matched-sample design can be referred to as
blocking.
This design often leads to a smaller sampling error
than the independent-sample design because
variation between sampled items is eliminated as a
source of sampling error.
31
Example: Express Deliveries
Inference About the Difference between the Means of
Two Populations: Matched Samples
A Chicago-based firm has documents that must
be quickly distributed to district offices throughout
the U.S. The firm must decide between two delivery
services, UPX (United Parcel Express) and INTEX
(International Express), to transport its documents.
In testing the delivery times of the two services, the
firm sent two reports to a random sample of ten
district offices with one report carried by UPX and
the other report carried by INTEX.
Do the data that follow indicate a difference in
mean delivery times for the two services?
32
Delivery Time (Hours)
District Office UPX INTEX Difference
Seattle 32 25 7
Los Angeles 30 24 6
Boston 19 15 4
Cleveland 16 15 1
New York 15 13 2
Houston 18 15 3
Atlanta 14 15 -1
St. Louis 10 8 2
Milwaukee 7 9 -2
Denver 16 11 5
Example: Express Deliveries
33
Inference About the Difference between the Means of
Two Populations: Matched Samples
Let µd= the mean of the difference values for the
two delivery services for the population of
district offices
– Hypotheses
H0: µd= 0, Ha: µd ≠ 0
Example: Express Deliveries
34
n Inference About the Difference between the Means of
Two Populations: Matched Samples
• Rejection Rule
Assuming the population of difference values is
approximately normally distributed, the t
distribution with n - 1 degrees of freedom applies.
With α = .05, t.025 = 2.262 (9 degrees of freedom).
Reject H0 if t < _________ or if t > __________
Example: Express DeliveriesExample: Express Deliveries
35
Inference About the Difference between the Means of
Two Populations: Matched Samples
d
d
n
i
=
∑
=
+ + +
=
( ... )
.
7 6 5
10
2 7
s
d d
n
d
i
=
−∑
−
= =
( ) .
.
2
1
76 1
9
2 9
t
d
s n
d
d
=
−
=
−
=
µ 2 7 0
2 9 10
2 94
.
.
.
Example: Express Deliveries
36
n Inference About the Difference between the Means of
Two Populations: Matched Samples
• Conclusion
Reject H0.
There is a significant difference between the mean
delivery times for the two services.
Example: Express DeliveriesExample: Express Deliveries
37
Introduction to Analysis of Variance
Analysis of Variance (ANOVA) can be used to test
for the equality of three or more population means
using data obtained from observational or
experimental studies.
We want to use the sample results to test the
following hypotheses.
H0: µ1 = µ2 = µ3 = . . .= µk
Ha: Not all population means are equal
38
Introduction to Analysis of Variance
n If H0 is rejected, we cannot conclude that all
population means are different.
n Rejecting H0 means that at least two population
means have different values.
39
Assumptions for Analysis of Variance
For each population, the response variable is normally
distributed.
The variance of the response variable, denoted σ 2
, is
the same for all of the populations.
The observations must be independent.
40
Analysis of Variance:
Testing for the Equality of k Population Means
Between-Treatments Estimate of Population Variance
Within-Treatments Estimate of Population Variance
Comparing the Variance Estimates: The F Test
The ANOVA Table
41
A between-treatment estimate of σ 2
is called the
mean square treatment and is denoted MSTR.
The numerator of MSTR is called the sum of squares
treatment and is denoted SSTR.
The denominator of MSTR represents the degrees of
freedom associated with SSTR.
Between-Treatments Estimate
of Population Variance
1
)(
MSTR
1
2
−
−
=
∑=
k
xxn
k
j
jj
42
The estimate of σ 2
based on the variation of the
sample observations within each sample is called the
mean square error and is denoted by MSE.
The numerator of MSE is called the sum of squares
error and is denoted by SSE.
The denominator of MSE represents the degrees of
freedom associated with SSE.
Within-Samples Estimate
of Population Variance
kn
sn
T
k
j
jj
−
−
=
∑=1
2
)1(
MSE
43
Comparing the Variance Estimates: The F Test
If the null hypothesis is true and the ANOVA
assumptions are valid, the sampling distribution of
MSTR/MSE is an F distribution with MSTR d.f. equal
to k - 1 and MSE d.f. equal to nT - k.
If the means of the k populations are not equal, the
value of MSTR/MSE will be inflated because MSTR
overestimates σ 2
.
Hence, we will reject H0 if the resulting value of
MSTR/MSE appears to be too large to have been
selected at random from the appropriate F
distribution.
44
Test for the Equality of k Population Means
Hypotheses
H0: µ1 = µ2 = µ3 = . . .= µk
Ha: Not all population means are equal
Test Statistic
F = MSTR/MSE
Rejection Rule
Reject H0 if F > Fα
where the value of Fα is based on an F distribution with
k - 1 numerator degrees of freedom and nT - 1
denominator degrees of freedom.
45
Sampling Distribution of MSTR/MSE
The figure below shows the rejection region associated
with a level of significance equal to α where Fα denotes
the critical value.
Do Not Reject H0 Reject H0
MSTR/MSE
Critical Value
Fα
46
ANOVA Table
Source of Sum of Degrees of Mean
Variation Squares Freedom Squares F
Treatment SSTR k - 1 MSTR MSTR/MSE
Error SSE nT- k MSE
Total SST nT - 1
SST divided by its degrees of freedom nT - 1 is simply
the overall sample variance that would be obtained if
we treated the entire nT observations as one data set.
∑∑= =
+=−=
k
j
n
i
ij
j
xx
1 1
2
SSESSTR)(SST
47
Example: Reed Manufacturing
Analysis of Variance
J. R. Reed would like to know if the mean
number of hours worked per week is the same for the
department managers at her three manufacturing
plants (Buffalo, Pittsburgh, and Detroit).
A simple random sample of 5 managers from
each of the three plants was taken and the number of
hours worked by each manager for the previous
week is shown on the next slide.
48
Analysis of Variance
Plant 1 Plant 2 Plant 3
Observation Buffalo Pittsburgh Detroit
1 48 73 51
2 54 63 63
3 57 66 61
4 54 64 54
5 62 74 56
Sample Mean 55 68 57
Sample Variance ____ _____ ______
Example: Reed Manufacturing
49
Analysis of Variance
– Hypotheses
H0: µ1 = µ2 = µ3
Ha: Not all the means are equal
where:
µ1 = mean number of hours worked per
week by the managers at Plant 1
µ2= mean number of hours worked per
week by the managers at Plant 2
µ3 = mean number of hours worked per
week by the managers at Plant 3
Example: Reed Manufacturing
50
Analysis of Variance
– Mean Square Treatment
Since the sample sizes are all equal
x = (55 + 68 + 57)/3 = ____
SSTR = 5(55 - 60)2
+ 5(68 - 60)2
+ 5(57 - 60)2
= ____
MSTR = 490/(3 - 1) = 245
– Mean Square Error
SSE = 4(26.0) + 4(26.5) + 4(24.5) = _____
MSE = 308/(15 - 3) = 25.667
=
Example: Reed Manufacturing
51
Analysis of Variance
– F - Test
If H0 is true, the ratio MSTR/MSE should be near
1 since both MSTR and MSE are estimating σ 2
. If
Hais true, the ratio should be significantly larger
than 1 since MSTR tends to overestimate σ 2
.
Example: Reed Manufacturing
52
n Analysis of Variance
•Rejection Rule
Assuming α = .05, F.05 = 3.89 (2 d.f. numerator,
12 d.f. denominator). Reject H0 if F > _______
•Test Statistic
F = MSTR/MSE = 245/25.667 = _______
Example: Reed ManufacturingExample: Reed Manufacturing
53
Analysis of Variance
– ANOVA Table
Source of Sum of Degrees of Mean
Variation Squares Freedom Square F
Treatments 490 2 245 9.55
Error 308 12 25.667
Total 798 14
Example: Reed Manufacturing
54
n Analysis of Variance
•Conclusion
F = 9.55 > F.05 = _____, so we reject H0. The mean
number of hours worked per week by department
managers is not the same at each plant.
Example: Reed ManufacturingExample: Reed Manufacturing
55
End of Chapter 10

More Related Content

What's hot

Chapter7
Chapter7Chapter7
Solution manual for design and analysis of experiments 9th edition douglas ...
Solution manual for design and analysis of experiments 9th edition   douglas ...Solution manual for design and analysis of experiments 9th edition   douglas ...
Solution manual for design and analysis of experiments 9th edition douglas ...
Salehkhanovic
 
Chapter4
Chapter4Chapter4
Chapter12
Chapter12Chapter12
Chapter12
Richard Ferreria
 
Chapter9
Chapter9Chapter9
Solutions. Design and Analysis of Experiments. Montgomery
Solutions. Design and Analysis of Experiments. MontgomerySolutions. Design and Analysis of Experiments. Montgomery
Solutions. Design and Analysis of Experiments. MontgomeryByron CZ
 
Chapter5
Chapter5Chapter5
Applied statistics and probability for engineers solution montgomery && runger
Applied statistics and probability for engineers solution   montgomery && rungerApplied statistics and probability for engineers solution   montgomery && runger
Applied statistics and probability for engineers solution montgomery && rungerAnkit Katiyar
 
Some Unbiased Classes of Estimators of Finite Population Mean
Some Unbiased Classes of Estimators of Finite Population MeanSome Unbiased Classes of Estimators of Finite Population Mean
Some Unbiased Classes of Estimators of Finite Population Mean
inventionjournals
 
Statistics assignment 8
Statistics assignment 8Statistics assignment 8
Statistics assignment 8
Ishaq Ahmed
 
Malhotra12
Malhotra12Malhotra12
Les5e ppt 04
Les5e ppt 04Les5e ppt 04
Les5e ppt 04
Subas Nandy
 
Discrete Probability Distributions.
Discrete Probability Distributions.Discrete Probability Distributions.
Discrete Probability Distributions.
ConflagratioNal Jahid
 
Decision analysis
Decision analysis Decision analysis
Decision analysis
Lili Fajri Dailimi
 

What's hot (15)

Chapter7
Chapter7Chapter7
Chapter7
 
Solution manual for design and analysis of experiments 9th edition douglas ...
Solution manual for design and analysis of experiments 9th edition   douglas ...Solution manual for design and analysis of experiments 9th edition   douglas ...
Solution manual for design and analysis of experiments 9th edition douglas ...
 
Chapter4
Chapter4Chapter4
Chapter4
 
Chapter12
Chapter12Chapter12
Chapter12
 
Chapter9
Chapter9Chapter9
Chapter9
 
Solutions. Design and Analysis of Experiments. Montgomery
Solutions. Design and Analysis of Experiments. MontgomerySolutions. Design and Analysis of Experiments. Montgomery
Solutions. Design and Analysis of Experiments. Montgomery
 
Chapter5
Chapter5Chapter5
Chapter5
 
S7 pn
S7 pnS7 pn
S7 pn
 
Applied statistics and probability for engineers solution montgomery && runger
Applied statistics and probability for engineers solution   montgomery && rungerApplied statistics and probability for engineers solution   montgomery && runger
Applied statistics and probability for engineers solution montgomery && runger
 
Some Unbiased Classes of Estimators of Finite Population Mean
Some Unbiased Classes of Estimators of Finite Population MeanSome Unbiased Classes of Estimators of Finite Population Mean
Some Unbiased Classes of Estimators of Finite Population Mean
 
Statistics assignment 8
Statistics assignment 8Statistics assignment 8
Statistics assignment 8
 
Malhotra12
Malhotra12Malhotra12
Malhotra12
 
Les5e ppt 04
Les5e ppt 04Les5e ppt 04
Les5e ppt 04
 
Discrete Probability Distributions.
Discrete Probability Distributions.Discrete Probability Distributions.
Discrete Probability Distributions.
 
Decision analysis
Decision analysis Decision analysis
Decision analysis
 

Viewers also liked

If this is the future, where is my tree of life?
If this is the future, where is my tree of life?If this is the future, where is my tree of life?
If this is the future, where is my tree of life?
Karen Cranston
 
Study on genetic diversity and phylogenetic relationship among eleven traditi...
Study on genetic diversity and phylogenetic relationship among eleven traditi...Study on genetic diversity and phylogenetic relationship among eleven traditi...
Study on genetic diversity and phylogenetic relationship among eleven traditi...
Manya Education Pvt Ltd
 
Phylogeny
PhylogenyPhylogeny
Phylogeny
martyynyyte
 
What is a phylogenetic tree
What is a phylogenetic treeWhat is a phylogenetic tree
What is a phylogenetic treeislam jan buneri
 
Phylogenetic trees
Phylogenetic treesPhylogenetic trees
Phylogenetic trees
martyynyyte
 

Viewers also liked (7)

If this is the future, where is my tree of life?
If this is the future, where is my tree of life?If this is the future, where is my tree of life?
If this is the future, where is my tree of life?
 
Study on genetic diversity and phylogenetic relationship among eleven traditi...
Study on genetic diversity and phylogenetic relationship among eleven traditi...Study on genetic diversity and phylogenetic relationship among eleven traditi...
Study on genetic diversity and phylogenetic relationship among eleven traditi...
 
Phylogeny
PhylogenyPhylogeny
Phylogeny
 
Phylogeny
PhylogenyPhylogeny
Phylogeny
 
What is a phylogenetic tree
What is a phylogenetic treeWhat is a phylogenetic tree
What is a phylogenetic tree
 
Phylogenetic analysis
Phylogenetic analysisPhylogenetic analysis
Phylogenetic analysis
 
Phylogenetic trees
Phylogenetic treesPhylogenetic trees
Phylogenetic trees
 

Similar to Kxu stat-anderson-ch10-student 2

Ch 10-Two populations.pptx
Ch 10-Two populations.pptxCh 10-Two populations.pptx
Ch 10-Two populations.pptx
tharkistani
 
Hypothesis testing - II.pptx
Hypothesis testing - II.pptxHypothesis testing - II.pptx
Hypothesis testing - II.pptx
ShashvatSingh12
 
Chapter07.pdf
Chapter07.pdfChapter07.pdf
Chapter07.pdf
KarenJoyBabida
 
Tables and Formulas for Sullivan, Statistics Informed Decisio.docx
Tables and Formulas for Sullivan, Statistics Informed Decisio.docxTables and Formulas for Sullivan, Statistics Informed Decisio.docx
Tables and Formulas for Sullivan, Statistics Informed Decisio.docx
mattinsonjanel
 
Estimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or VarianceEstimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or Variance
Long Beach City College
 
Estimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or Variance Estimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or Variance
Long Beach City College
 
Chapter one on sampling distributions.ppt
Chapter one on sampling distributions.pptChapter one on sampling distributions.ppt
Chapter one on sampling distributions.ppt
FekaduAman
 
Wilcoxon Rank-Sum Test
Wilcoxon Rank-Sum TestWilcoxon Rank-Sum Test
Wilcoxon Rank-Sum Test
Sahil Jain
 
Estimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or VarianceEstimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or Variance
Long Beach City College
 
Estimation by c.i
Estimation by c.iEstimation by c.i
Estimation by c.i
Shahab Yaseen
 
Statistik 1 6 distribusi probabilitas normal
Statistik 1 6 distribusi probabilitas normalStatistik 1 6 distribusi probabilitas normal
Statistik 1 6 distribusi probabilitas normalSelvin Hadi
 
Sampling Distribution -I
Sampling Distribution -ISampling Distribution -I
Sampling Distribution -ISadam Hussen
 
STATISTICS-MODULE 2.MEAN OF DISCRETE RANDOM VARIABLE
STATISTICS-MODULE 2.MEAN OF DISCRETE RANDOM VARIABLESTATISTICS-MODULE 2.MEAN OF DISCRETE RANDOM VARIABLE
STATISTICS-MODULE 2.MEAN OF DISCRETE RANDOM VARIABLE
joycelatombo0419
 
Mean, median, and mode ug
Mean, median, and mode ugMean, median, and mode ug
Mean, median, and mode ugAbhishekDas15
 
Hypothesis Test _Two-sample t-test, Z-test, Proportion Z-test
Hypothesis Test _Two-sample t-test, Z-test, Proportion Z-testHypothesis Test _Two-sample t-test, Z-test, Proportion Z-test
Hypothesis Test _Two-sample t-test, Z-test, Proportion Z-test
Ravindra Nath Shukla
 
t-z-chi-square tests of sig.pdf
t-z-chi-square tests of sig.pdft-z-chi-square tests of sig.pdf
t-z-chi-square tests of sig.pdf
AmoghLavania1
 
Two Means, Independent Samples
Two Means, Independent SamplesTwo Means, Independent Samples
Two Means, Independent Samples
Long Beach City College
 
Session 03 Probability & sampling Distribution NEW.pptx
Session 03 Probability & sampling Distribution NEW.pptxSession 03 Probability & sampling Distribution NEW.pptx
Session 03 Probability & sampling Distribution NEW.pptx
Muneer Akhter
 

Similar to Kxu stat-anderson-ch10-student 2 (20)

Ch 10-Two populations.pptx
Ch 10-Two populations.pptxCh 10-Two populations.pptx
Ch 10-Two populations.pptx
 
Hypothesis testing - II.pptx
Hypothesis testing - II.pptxHypothesis testing - II.pptx
Hypothesis testing - II.pptx
 
Chapter07.pdf
Chapter07.pdfChapter07.pdf
Chapter07.pdf
 
Tables and Formulas for Sullivan, Statistics Informed Decisio.docx
Tables and Formulas for Sullivan, Statistics Informed Decisio.docxTables and Formulas for Sullivan, Statistics Informed Decisio.docx
Tables and Formulas for Sullivan, Statistics Informed Decisio.docx
 
Student t t est
Student t t estStudent t t est
Student t t est
 
Estimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or VarianceEstimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or Variance
 
Estimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or Variance Estimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or Variance
 
Chapter one on sampling distributions.ppt
Chapter one on sampling distributions.pptChapter one on sampling distributions.ppt
Chapter one on sampling distributions.ppt
 
Wilcoxon Rank-Sum Test
Wilcoxon Rank-Sum TestWilcoxon Rank-Sum Test
Wilcoxon Rank-Sum Test
 
Estimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or VarianceEstimating a Population Standard Deviation or Variance
Estimating a Population Standard Deviation or Variance
 
Estimation by c.i
Estimation by c.iEstimation by c.i
Estimation by c.i
 
Statistik 1 6 distribusi probabilitas normal
Statistik 1 6 distribusi probabilitas normalStatistik 1 6 distribusi probabilitas normal
Statistik 1 6 distribusi probabilitas normal
 
Sampling Distribution -I
Sampling Distribution -ISampling Distribution -I
Sampling Distribution -I
 
STATISTICS-MODULE 2.MEAN OF DISCRETE RANDOM VARIABLE
STATISTICS-MODULE 2.MEAN OF DISCRETE RANDOM VARIABLESTATISTICS-MODULE 2.MEAN OF DISCRETE RANDOM VARIABLE
STATISTICS-MODULE 2.MEAN OF DISCRETE RANDOM VARIABLE
 
Statistical ppt
Statistical pptStatistical ppt
Statistical ppt
 
Mean, median, and mode ug
Mean, median, and mode ugMean, median, and mode ug
Mean, median, and mode ug
 
Hypothesis Test _Two-sample t-test, Z-test, Proportion Z-test
Hypothesis Test _Two-sample t-test, Z-test, Proportion Z-testHypothesis Test _Two-sample t-test, Z-test, Proportion Z-test
Hypothesis Test _Two-sample t-test, Z-test, Proportion Z-test
 
t-z-chi-square tests of sig.pdf
t-z-chi-square tests of sig.pdft-z-chi-square tests of sig.pdf
t-z-chi-square tests of sig.pdf
 
Two Means, Independent Samples
Two Means, Independent SamplesTwo Means, Independent Samples
Two Means, Independent Samples
 
Session 03 Probability & sampling Distribution NEW.pptx
Session 03 Probability & sampling Distribution NEW.pptxSession 03 Probability & sampling Distribution NEW.pptx
Session 03 Probability & sampling Distribution NEW.pptx
 

Recently uploaded

Search Disrupted Google’s Leaked Documents Rock the SEO World.pdf
Search Disrupted Google’s Leaked Documents Rock the SEO World.pdfSearch Disrupted Google’s Leaked Documents Rock the SEO World.pdf
Search Disrupted Google’s Leaked Documents Rock the SEO World.pdf
Arihant Webtech Pvt. Ltd
 
3.0 Project 2_ Developing My Brand Identity Kit.pptx
3.0 Project 2_ Developing My Brand Identity Kit.pptx3.0 Project 2_ Developing My Brand Identity Kit.pptx
3.0 Project 2_ Developing My Brand Identity Kit.pptx
tanyjahb
 
Cree_Rey_BrandIdentityKit.PDF_PersonalBd
Cree_Rey_BrandIdentityKit.PDF_PersonalBdCree_Rey_BrandIdentityKit.PDF_PersonalBd
Cree_Rey_BrandIdentityKit.PDF_PersonalBd
creerey
 
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s DholeraTata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Avirahi City Dholera
 
Brand Analysis for an artist named Struan
Brand Analysis for an artist named StruanBrand Analysis for an artist named Struan
Brand Analysis for an artist named Struan
sarahvanessa51503
 
ikea_woodgreen_petscharity_dog-alogue_digital.pdf
ikea_woodgreen_petscharity_dog-alogue_digital.pdfikea_woodgreen_petscharity_dog-alogue_digital.pdf
ikea_woodgreen_petscharity_dog-alogue_digital.pdf
agatadrynko
 
The Parable of the Pipeline a book every new businessman or business student ...
The Parable of the Pipeline a book every new businessman or business student ...The Parable of the Pipeline a book every new businessman or business student ...
The Parable of the Pipeline a book every new businessman or business student ...
awaisafdar
 
Memorandum Of Association Constitution of Company.ppt
Memorandum Of Association Constitution of Company.pptMemorandum Of Association Constitution of Company.ppt
Memorandum Of Association Constitution of Company.ppt
seri bangash
 
VAT Registration Outlined In UAE: Benefits and Requirements
VAT Registration Outlined In UAE: Benefits and RequirementsVAT Registration Outlined In UAE: Benefits and Requirements
VAT Registration Outlined In UAE: Benefits and Requirements
uae taxgpt
 
Sustainability: Balancing the Environment, Equity & Economy
Sustainability: Balancing the Environment, Equity & EconomySustainability: Balancing the Environment, Equity & Economy
Sustainability: Balancing the Environment, Equity & Economy
Operational Excellence Consulting
 
Project File Report BBA 6th semester.pdf
Project File Report BBA 6th semester.pdfProject File Report BBA 6th semester.pdf
Project File Report BBA 6th semester.pdf
RajPriye
 
Attending a job Interview for B1 and B2 Englsih learners
Attending a job Interview for B1 and B2 Englsih learnersAttending a job Interview for B1 and B2 Englsih learners
Attending a job Interview for B1 and B2 Englsih learners
Erika906060
 
Maksym Vyshnivetskyi: PMO Quality Management (UA)
Maksym Vyshnivetskyi: PMO Quality Management (UA)Maksym Vyshnivetskyi: PMO Quality Management (UA)
Maksym Vyshnivetskyi: PMO Quality Management (UA)
Lviv Startup Club
 
Enterprise Excellence is Inclusive Excellence.pdf
Enterprise Excellence is Inclusive Excellence.pdfEnterprise Excellence is Inclusive Excellence.pdf
Enterprise Excellence is Inclusive Excellence.pdf
KaiNexus
 
BeMetals Presentation_May_22_2024 .pdf
BeMetals Presentation_May_22_2024   .pdfBeMetals Presentation_May_22_2024   .pdf
BeMetals Presentation_May_22_2024 .pdf
DerekIwanaka1
 
Putting the SPARK into Virtual Training.pptx
Putting the SPARK into Virtual Training.pptxPutting the SPARK into Virtual Training.pptx
Putting the SPARK into Virtual Training.pptx
Cynthia Clay
 
Buy Verified PayPal Account | Buy Google 5 Star Reviews
Buy Verified PayPal Account | Buy Google 5 Star ReviewsBuy Verified PayPal Account | Buy Google 5 Star Reviews
Buy Verified PayPal Account | Buy Google 5 Star Reviews
usawebmarket
 
Discover the innovative and creative projects that highlight my journey throu...
Discover the innovative and creative projects that highlight my journey throu...Discover the innovative and creative projects that highlight my journey throu...
Discover the innovative and creative projects that highlight my journey throu...
dylandmeas
 
ModelingMarketingStrategiesMKS.CollumbiaUniversitypdf
ModelingMarketingStrategiesMKS.CollumbiaUniversitypdfModelingMarketingStrategiesMKS.CollumbiaUniversitypdf
ModelingMarketingStrategiesMKS.CollumbiaUniversitypdf
fisherameliaisabella
 
amptalk_RecruitingDeck_english_2024.06.05
amptalk_RecruitingDeck_english_2024.06.05amptalk_RecruitingDeck_english_2024.06.05
amptalk_RecruitingDeck_english_2024.06.05
marketing317746
 

Recently uploaded (20)

Search Disrupted Google’s Leaked Documents Rock the SEO World.pdf
Search Disrupted Google’s Leaked Documents Rock the SEO World.pdfSearch Disrupted Google’s Leaked Documents Rock the SEO World.pdf
Search Disrupted Google’s Leaked Documents Rock the SEO World.pdf
 
3.0 Project 2_ Developing My Brand Identity Kit.pptx
3.0 Project 2_ Developing My Brand Identity Kit.pptx3.0 Project 2_ Developing My Brand Identity Kit.pptx
3.0 Project 2_ Developing My Brand Identity Kit.pptx
 
Cree_Rey_BrandIdentityKit.PDF_PersonalBd
Cree_Rey_BrandIdentityKit.PDF_PersonalBdCree_Rey_BrandIdentityKit.PDF_PersonalBd
Cree_Rey_BrandIdentityKit.PDF_PersonalBd
 
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s DholeraTata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
Tata Group Dials Taiwan for Its Chipmaking Ambition in Gujarat’s Dholera
 
Brand Analysis for an artist named Struan
Brand Analysis for an artist named StruanBrand Analysis for an artist named Struan
Brand Analysis for an artist named Struan
 
ikea_woodgreen_petscharity_dog-alogue_digital.pdf
ikea_woodgreen_petscharity_dog-alogue_digital.pdfikea_woodgreen_petscharity_dog-alogue_digital.pdf
ikea_woodgreen_petscharity_dog-alogue_digital.pdf
 
The Parable of the Pipeline a book every new businessman or business student ...
The Parable of the Pipeline a book every new businessman or business student ...The Parable of the Pipeline a book every new businessman or business student ...
The Parable of the Pipeline a book every new businessman or business student ...
 
Memorandum Of Association Constitution of Company.ppt
Memorandum Of Association Constitution of Company.pptMemorandum Of Association Constitution of Company.ppt
Memorandum Of Association Constitution of Company.ppt
 
VAT Registration Outlined In UAE: Benefits and Requirements
VAT Registration Outlined In UAE: Benefits and RequirementsVAT Registration Outlined In UAE: Benefits and Requirements
VAT Registration Outlined In UAE: Benefits and Requirements
 
Sustainability: Balancing the Environment, Equity & Economy
Sustainability: Balancing the Environment, Equity & EconomySustainability: Balancing the Environment, Equity & Economy
Sustainability: Balancing the Environment, Equity & Economy
 
Project File Report BBA 6th semester.pdf
Project File Report BBA 6th semester.pdfProject File Report BBA 6th semester.pdf
Project File Report BBA 6th semester.pdf
 
Attending a job Interview for B1 and B2 Englsih learners
Attending a job Interview for B1 and B2 Englsih learnersAttending a job Interview for B1 and B2 Englsih learners
Attending a job Interview for B1 and B2 Englsih learners
 
Maksym Vyshnivetskyi: PMO Quality Management (UA)
Maksym Vyshnivetskyi: PMO Quality Management (UA)Maksym Vyshnivetskyi: PMO Quality Management (UA)
Maksym Vyshnivetskyi: PMO Quality Management (UA)
 
Enterprise Excellence is Inclusive Excellence.pdf
Enterprise Excellence is Inclusive Excellence.pdfEnterprise Excellence is Inclusive Excellence.pdf
Enterprise Excellence is Inclusive Excellence.pdf
 
BeMetals Presentation_May_22_2024 .pdf
BeMetals Presentation_May_22_2024   .pdfBeMetals Presentation_May_22_2024   .pdf
BeMetals Presentation_May_22_2024 .pdf
 
Putting the SPARK into Virtual Training.pptx
Putting the SPARK into Virtual Training.pptxPutting the SPARK into Virtual Training.pptx
Putting the SPARK into Virtual Training.pptx
 
Buy Verified PayPal Account | Buy Google 5 Star Reviews
Buy Verified PayPal Account | Buy Google 5 Star ReviewsBuy Verified PayPal Account | Buy Google 5 Star Reviews
Buy Verified PayPal Account | Buy Google 5 Star Reviews
 
Discover the innovative and creative projects that highlight my journey throu...
Discover the innovative and creative projects that highlight my journey throu...Discover the innovative and creative projects that highlight my journey throu...
Discover the innovative and creative projects that highlight my journey throu...
 
ModelingMarketingStrategiesMKS.CollumbiaUniversitypdf
ModelingMarketingStrategiesMKS.CollumbiaUniversitypdfModelingMarketingStrategiesMKS.CollumbiaUniversitypdf
ModelingMarketingStrategiesMKS.CollumbiaUniversitypdf
 
amptalk_RecruitingDeck_english_2024.06.05
amptalk_RecruitingDeck_english_2024.06.05amptalk_RecruitingDeck_english_2024.06.05
amptalk_RecruitingDeck_english_2024.06.05
 

Kxu stat-anderson-ch10-student 2

  • 1. 1 Chapter 10 Comparisons Involving Means µµ11 == µµ22 ?? ANOVAANOVA Estimation of the Difference between the Means of Two Populations: Independent Samples Hypothesis Tests about the Difference between the Means of Two Populations: Independent Samples Inferences about the Difference between the Means of Two Populations: Matched Samples Introduction to Analysis of Variance (ANOVA) ANOVA: Testing for the Equality of k Population Means
  • 2. 2 Estimation of the Difference Between the Means of Two Populations: Independent Samples Point Estimator of the Difference between the Means of Two Populations Sampling Distribution Interval Estimate of µ1−µ2: Large-Sample Case Interval Estimate of µ1−µ2: Small-Sample Case x x1 2−
  • 3. 3 Point Estimator of the Difference Between the Means of Two Populations Let µ1 equal the mean of population 1 and µ2 equal the mean of population 2. The difference between the two population means is µ1 - µ2. To estimate µ1 - µ2, we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2. Let equal the mean of sample 1 and equal the mean of sample 2. The point estimator of the difference between the means of the populations 1 and 2 is . x x1 2− x1 x2
  • 4. 4 E x x( )1 2 1 2− = −µ µ n Properties of the Sampling Distribution ofProperties of the Sampling Distribution of • Expected ValueExpected Value Sampling Distribution ofSampling Distribution of x x1 2− x x1 2−
  • 5. 5 Properties of the Sampling Distribution of – Standard Deviation where: σ1 = standard deviation of population 1 σ2 = standard deviation of population 2 n1= sample size from population 1 n2 = sample size from population 2 Sampling Distribution of x x1 2− x x1 2− σ σ σ x x n n1 2 1 2 1 2 2 2 − = +
  • 6. 6 Interval Estimate with σ1 and σ2 Known where: 1 - α is the confidence coefficient (level). Interval Estimate of µ1 - µ2: Large-Sample Case (n1 > 30 and n2 > 30) x x z x x1 2 2 1 2 − ± −α σ/
  • 7. 7 n Interval Estimate withInterval Estimate with σσ11 andand σσ22 UnknownUnknown where:where: Interval Estimate of µ1 - µ2: Large-Sample Case (n1 > 30 and n2 > 30) x x z sx x1 2 2 1 2 − ± −α/ s s n s n x x1 2 1 2 1 2 2 2 − = +
  • 8. 8 Example: Par, Inc. Interval Estimate of µ1 - µ2: Large-Sample Case Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.” In a test of driving distance using a mechanical driving device, a sample of Par golf balls was compared with a sample of golf balls made by Rap, Ltd., a competitor. The sample statistics appear on the next slide.
  • 9. 9 Example: Par, Inc. Interval Estimate of µ1 - µ2: Large-Sample Case – Sample Statistics Sample #1 Sample #2 Par, Inc. Rap, Ltd. Sample Size n1 = 120 balls n2 = 80 balls Mean = 235 yards = 218 yards Standard Dev. s1 = ___yards s2 =____ yards x1 2x
  • 10. 10 Point Estimate of the Difference Between Two Population Means µ1 = mean distance for the population of Par, Inc. golf balls µ2 = mean distance for the population of Rap, Ltd. golf balls Point estimate of µ1 - µ2 = = 235 - 218 = 17 yards.x x1 2− Example: Par, Inc.
  • 11. 11 Point Estimator of the Difference Between the Means of Two Populations Population 1Population 1 Par, Inc. Golf BallsPar, Inc. Golf Balls µµ11 = mean driving= mean driving distance of Pardistance of Par golf ballsgolf balls Population 1Population 1 Par, Inc. Golf BallsPar, Inc. Golf Balls µµ11 = mean driving= mean driving distance of Pardistance of Par golf ballsgolf balls Population 2Population 2 Rap, Ltd. Golf BallsRap, Ltd. Golf Balls µµ22 = mean driving= mean driving distance of Rapdistance of Rap golf ballsgolf balls Population 2Population 2 Rap, Ltd. Golf BallsRap, Ltd. Golf Balls µµ22 = mean driving= mean driving distance of Rapdistance of Rap golf ballsgolf balls µµ11 –– µµ22 = difference between= difference between the mean distancesthe mean distances Simple random sampleSimple random sample ofof nn11 Par golf ballsPar golf balls xx11 = sample mean distance= sample mean distance for sample of Par golf ballfor sample of Par golf ball Simple random sampleSimple random sample ofof nn11 Par golf ballsPar golf balls xx11 = sample mean distance= sample mean distance for sample of Par golf ballfor sample of Par golf ball Simple random sampleSimple random sample ofof nn22 Rap golf ballsRap golf balls xx22 = sample mean distance= sample mean distance for sample of Rap golf ballfor sample of Rap golf ball Simple random sampleSimple random sample ofof nn22 Rap golf ballsRap golf balls xx22 = sample mean distance= sample mean distance for sample of Rap golf ballfor sample of Rap golf ball xx11 -- xx22 = Point Estimate of= Point Estimate of µµ11 –– µµ22
  • 12. 12 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, σ1 and σ2 Unknown Substituting the sample standard deviations for the population standard deviation: = ___________ or 11.86 yards to 22.14 yards. We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and Rap, Ltd. balls lies in the interval of _______________ yards. x x z n n 1 2 2 1 2 1 2 2 2 2 2 17 1 96 15 120 20 80 − ± + = ± +α σ σ / . ( ) ( ) Example: Par, Inc.
  • 13. 13 Interval Estimate of µ1 - µ2: Small-Sample Case (n1 < 30 and/or n2 < 30) Interval Estimate with σ 2 Known (and equal) where: x x z x x1 2 2 1 2 − ± −α σ/ σ σx x n n1 2 2 1 2 1 1 − = +( )
  • 14. 14 Interval Estimate with σ 2 Unknown (and assumed equal) where: and the degrees of freedom for the t-distribution is n1+n2-2. Interval Estimate of µ1 - µ2: Small-Sample Case (n1 < 30 and/or n2 < 30) x x t sx x1 2 2 1 2 − ± −α / s n s n s n n 2 1 1 2 2 2 2 1 2 1 1 2 = − + − + − ( ) ( ) s s n n x x1 2 2 1 2 1 1 − = +( )
  • 15. 15 Example: Specific Motors Specific Motors of Detroit has developed a new automobile known as the M car. 12 M cars and 8 J cars (from Japan) were road tested to compare miles-per- gallon (mpg) performance. The sample statistics are: Sample #1 Sample #2 M Cars J Cars Sample Size n1 = 12 cars n2 = 8 cars Mean = 29.8 mpg = 27.3 mpg Standard Deviation s1 = ____ mpg s2 = ____ mpg x2x1
  • 16. 16 Point Estimate of the Difference Between Two Population Means µ1 = mean miles-per-gallon for the population of M cars µ2 = mean miles-per-gallon for the population of J cars Point estimate of µ1 - µ2 = = ________ = ___ mpg.x x1 2− Example: Specific Motors
  • 17. 17 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case We will make the following assumptions: – The miles per gallon rating must be normally distributed for both the M car and the J car. – The variance in the miles per gallon rating must be the same for both the M car and the J car. Example: Specific Motors
  • 18. 18 n 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case Using the t distribution with n1 + n2 - 2 = ___ degrees of freedom, the appropriate t value is t.025 = ______. We will use a weighted average of the two sample variances as the pooled estimator of σ 2 . Example: Specific Motors
  • 19. 19 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Case = _____________, or .3 to 4.7 miles per gallon. We are 95% confident that the difference between the mean mpg ratings of the two car types is from .3 to 4.7 mpg (with the M car having the higher mpg). s n s n s n n 2 1 1 2 2 2 2 1 2 2 2 1 1 2 11 2 56 7 1 81 12 8 2 5 28= − + − + − = + + − = ( ) ( ) ( . ) ( . ) . x x t s n n 1 2 025 2 1 2 1 1 2 5 2 101 5 28 1 12 1 8 − ± + = ± +. ( ) . . . ( ) Example: Specific Motors
  • 20. 20 Hypotheses H0: µ1- µ2 < 0 H0: µ1 - µ2 > 0 H0: µ1- µ2 = 0 Ha: µ1- µ2 > 0 Ha: µ1 - µ2 < 0 Ha: µ1- µ2 ≠ 0 Test Statistic Large-Sample Small-Sample Hypothesis Tests About the Difference between the Means of Two Populations: Independent Samples z x x n n = − − − + ( ) ( )1 2 1 2 1 2 1 2 2 2 µ µ σ σ t x x s n n = − − − + ( ) ( ) ( ) 1 2 1 2 2 1 21 1 µ µ
  • 21. 21 Hypothesis Tests About the Difference between the Means of Two Populations: Large-Sample Case Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.” In a test of driving distance using a mechanical driving device, a sample of Par golf balls was compared with a sample of golf balls made by Rap, Ltd., a competitor. The sample statistics appear on the next slide. Example: Par, Inc.
  • 22. 22 Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case – Sample Statistics Sample #1 Sample #2 Par, Inc. Rap, Ltd. Sample Size n1 = 120 balls n2 = 80 balls Mean = 235 yards = 218 yards Standard Dev. s1 = ____ yards s2 = ____ yards Example: Par, Inc. x1 x2
  • 23. 23 Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case Can we conclude, using a .01 level of significance, that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls? Example: Par, Inc.
  • 24. 24 n Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case µ1 = mean distance for the population of Par, Inc. golf balls µ2 = mean distance for the population of Rap, Ltd. golf balls •Hypotheses H0: µ1 - µ2 < 0 Ha: µ1 - µ2 > 0 Example: Par, Inc.Example: Par, Inc.
  • 25. 25 Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case – Rejection Rule Reject H0 if z > ________ z x x n n = − − − + = − − + = = ( ) ( ) ( ) ( ) ( ) . .1 2 1 2 1 2 1 2 2 2 2 2 235 218 0 15 120 20 80 17 2 62 6 49 µ µ σ σ Example: Par, Inc.
  • 26. 26 n Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample Case • Conclusion Reject H0. We are at least 99% confident that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls. Example: Par, Inc.Example: Par, Inc.
  • 27. 27 Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case Can we conclude, using a .05 level of significance, that the miles-per-gallon (mpg) performance of M cars is greater than the miles-per- gallon performance of J cars? Example: Specific Motors
  • 28. 28 n Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case µ1 = mean mpg for the population of M cars µ2 = mean mpg for the population of J cars •Hypotheses H0: µ1 - µ2 < 0 Ha: µ1 - µ2 > 0 Example: Specific MotorsExample: Specific Motors
  • 29. 29 Example: Specific Motors Hypothesis Tests About the Difference Between the Means of Two Populations: Small-Sample Case – Rejection Rule Reject H0 if t > _______ (a = .05, d.f. = 18) – Test Statistic where: t x x s n n = − − − + ( ) ( ) ( ) 1 2 1 2 2 1 21 1 µ µ 2 2 2 1 1 2 2 1 2 ( 1) ( 1) 2 n s n s s n n − + − = + −
  • 30. 30 Inference About the Difference between the Means of Two Populations: Matched Samples With a matched-sample design each sampled item provides a pair of data values. The matched-sample design can be referred to as blocking. This design often leads to a smaller sampling error than the independent-sample design because variation between sampled items is eliminated as a source of sampling error.
  • 31. 31 Example: Express Deliveries Inference About the Difference between the Means of Two Populations: Matched Samples A Chicago-based firm has documents that must be quickly distributed to district offices throughout the U.S. The firm must decide between two delivery services, UPX (United Parcel Express) and INTEX (International Express), to transport its documents. In testing the delivery times of the two services, the firm sent two reports to a random sample of ten district offices with one report carried by UPX and the other report carried by INTEX. Do the data that follow indicate a difference in mean delivery times for the two services?
  • 32. 32 Delivery Time (Hours) District Office UPX INTEX Difference Seattle 32 25 7 Los Angeles 30 24 6 Boston 19 15 4 Cleveland 16 15 1 New York 15 13 2 Houston 18 15 3 Atlanta 14 15 -1 St. Louis 10 8 2 Milwaukee 7 9 -2 Denver 16 11 5 Example: Express Deliveries
  • 33. 33 Inference About the Difference between the Means of Two Populations: Matched Samples Let µd= the mean of the difference values for the two delivery services for the population of district offices – Hypotheses H0: µd= 0, Ha: µd ≠ 0 Example: Express Deliveries
  • 34. 34 n Inference About the Difference between the Means of Two Populations: Matched Samples • Rejection Rule Assuming the population of difference values is approximately normally distributed, the t distribution with n - 1 degrees of freedom applies. With α = .05, t.025 = 2.262 (9 degrees of freedom). Reject H0 if t < _________ or if t > __________ Example: Express DeliveriesExample: Express Deliveries
  • 35. 35 Inference About the Difference between the Means of Two Populations: Matched Samples d d n i = ∑ = + + + = ( ... ) . 7 6 5 10 2 7 s d d n d i = −∑ − = = ( ) . . 2 1 76 1 9 2 9 t d s n d d = − = − = µ 2 7 0 2 9 10 2 94 . . . Example: Express Deliveries
  • 36. 36 n Inference About the Difference between the Means of Two Populations: Matched Samples • Conclusion Reject H0. There is a significant difference between the mean delivery times for the two services. Example: Express DeliveriesExample: Express Deliveries
  • 37. 37 Introduction to Analysis of Variance Analysis of Variance (ANOVA) can be used to test for the equality of three or more population means using data obtained from observational or experimental studies. We want to use the sample results to test the following hypotheses. H0: µ1 = µ2 = µ3 = . . .= µk Ha: Not all population means are equal
  • 38. 38 Introduction to Analysis of Variance n If H0 is rejected, we cannot conclude that all population means are different. n Rejecting H0 means that at least two population means have different values.
  • 39. 39 Assumptions for Analysis of Variance For each population, the response variable is normally distributed. The variance of the response variable, denoted σ 2 , is the same for all of the populations. The observations must be independent.
  • 40. 40 Analysis of Variance: Testing for the Equality of k Population Means Between-Treatments Estimate of Population Variance Within-Treatments Estimate of Population Variance Comparing the Variance Estimates: The F Test The ANOVA Table
  • 41. 41 A between-treatment estimate of σ 2 is called the mean square treatment and is denoted MSTR. The numerator of MSTR is called the sum of squares treatment and is denoted SSTR. The denominator of MSTR represents the degrees of freedom associated with SSTR. Between-Treatments Estimate of Population Variance 1 )( MSTR 1 2 − − = ∑= k xxn k j jj
  • 42. 42 The estimate of σ 2 based on the variation of the sample observations within each sample is called the mean square error and is denoted by MSE. The numerator of MSE is called the sum of squares error and is denoted by SSE. The denominator of MSE represents the degrees of freedom associated with SSE. Within-Samples Estimate of Population Variance kn sn T k j jj − − = ∑=1 2 )1( MSE
  • 43. 43 Comparing the Variance Estimates: The F Test If the null hypothesis is true and the ANOVA assumptions are valid, the sampling distribution of MSTR/MSE is an F distribution with MSTR d.f. equal to k - 1 and MSE d.f. equal to nT - k. If the means of the k populations are not equal, the value of MSTR/MSE will be inflated because MSTR overestimates σ 2 . Hence, we will reject H0 if the resulting value of MSTR/MSE appears to be too large to have been selected at random from the appropriate F distribution.
  • 44. 44 Test for the Equality of k Population Means Hypotheses H0: µ1 = µ2 = µ3 = . . .= µk Ha: Not all population means are equal Test Statistic F = MSTR/MSE Rejection Rule Reject H0 if F > Fα where the value of Fα is based on an F distribution with k - 1 numerator degrees of freedom and nT - 1 denominator degrees of freedom.
  • 45. 45 Sampling Distribution of MSTR/MSE The figure below shows the rejection region associated with a level of significance equal to α where Fα denotes the critical value. Do Not Reject H0 Reject H0 MSTR/MSE Critical Value Fα
  • 46. 46 ANOVA Table Source of Sum of Degrees of Mean Variation Squares Freedom Squares F Treatment SSTR k - 1 MSTR MSTR/MSE Error SSE nT- k MSE Total SST nT - 1 SST divided by its degrees of freedom nT - 1 is simply the overall sample variance that would be obtained if we treated the entire nT observations as one data set. ∑∑= = +=−= k j n i ij j xx 1 1 2 SSESSTR)(SST
  • 47. 47 Example: Reed Manufacturing Analysis of Variance J. R. Reed would like to know if the mean number of hours worked per week is the same for the department managers at her three manufacturing plants (Buffalo, Pittsburgh, and Detroit). A simple random sample of 5 managers from each of the three plants was taken and the number of hours worked by each manager for the previous week is shown on the next slide.
  • 48. 48 Analysis of Variance Plant 1 Plant 2 Plant 3 Observation Buffalo Pittsburgh Detroit 1 48 73 51 2 54 63 63 3 57 66 61 4 54 64 54 5 62 74 56 Sample Mean 55 68 57 Sample Variance ____ _____ ______ Example: Reed Manufacturing
  • 49. 49 Analysis of Variance – Hypotheses H0: µ1 = µ2 = µ3 Ha: Not all the means are equal where: µ1 = mean number of hours worked per week by the managers at Plant 1 µ2= mean number of hours worked per week by the managers at Plant 2 µ3 = mean number of hours worked per week by the managers at Plant 3 Example: Reed Manufacturing
  • 50. 50 Analysis of Variance – Mean Square Treatment Since the sample sizes are all equal x = (55 + 68 + 57)/3 = ____ SSTR = 5(55 - 60)2 + 5(68 - 60)2 + 5(57 - 60)2 = ____ MSTR = 490/(3 - 1) = 245 – Mean Square Error SSE = 4(26.0) + 4(26.5) + 4(24.5) = _____ MSE = 308/(15 - 3) = 25.667 = Example: Reed Manufacturing
  • 51. 51 Analysis of Variance – F - Test If H0 is true, the ratio MSTR/MSE should be near 1 since both MSTR and MSE are estimating σ 2 . If Hais true, the ratio should be significantly larger than 1 since MSTR tends to overestimate σ 2 . Example: Reed Manufacturing
  • 52. 52 n Analysis of Variance •Rejection Rule Assuming α = .05, F.05 = 3.89 (2 d.f. numerator, 12 d.f. denominator). Reject H0 if F > _______ •Test Statistic F = MSTR/MSE = 245/25.667 = _______ Example: Reed ManufacturingExample: Reed Manufacturing
  • 53. 53 Analysis of Variance – ANOVA Table Source of Sum of Degrees of Mean Variation Squares Freedom Square F Treatments 490 2 245 9.55 Error 308 12 25.667 Total 798 14 Example: Reed Manufacturing
  • 54. 54 n Analysis of Variance •Conclusion F = 9.55 > F.05 = _____, so we reject H0. The mean number of hours worked per week by department managers is not the same at each plant. Example: Reed ManufacturingExample: Reed Manufacturing