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Statistical
Inferences about Two
Populations
Learning Objectives
• Test hypotheses and construct confidenceintervals about the difference in two populationmeans using the Z statistic.
• Test hypotheses and construct confidenceintervals about the difference in two populationmeans using the t statistic.
QT-II/Ses 3/SAMPRIT
Learning Objectives
• Test hypotheses and construct confidence intervals about the differences in two population proportions.
• Test hypotheses and construct confidence intervals about two population variances.
QT-II/Ses 3/SAMPRIT
Sampling Distribution of the Difference Between
Two Sample Means
nx
x
1
1
Population 1
Population 2
nx
x
2
2
1 2X X
1X
2X
1 2X X
1x
1x1x 2x
2x
2x
QT-II/Ses 3/SAMPRIT
Sampling Distribution of the Difference
between Two Sample Means
1 2X X1 2X X
1 2
1
2
1
2
2
2X X n n
1 21 2X X 2121 xx
2
2
2
1
2
1
21 nnxx
21 xx21 xx
QT-II/Ses 3/SAMPRIT
Z Formula for the Difference
in Two Sample Means
nn
xxz
2
2
2
1
2
1
2121
When 12 and 2
2 are known and Independent Samples
QT-II/Ses 3/SAMPRIT
Hypothesis Testing for Differences Between
Means: Example with CI 95%
1 2X X
Rejection
Region
Non Rejection Region
Critical Values
Rejection
Region
1 2X X
025.2
025.2
H
H
o
a
:
:
1 2
1 2
0
0
21 xx 21 xx
QT-II/Ses 3/SAMPRIT
Hypothesis Testing for Differences Between
Means: The Wage Example (part 2)
.Hz
.Hzz
o
o
reject not do 1.96, 1.96- If
reject 1.96, > or 1.96- < IfRejection
Region
Non Rejection Region
Critical Values
Rejection
Region
96.1Z c0 96.1Z c
025.2
025.2
96.1cz96.1cz
QT-II/Ses 3/SAMPRIT
Example
• We want to conduct a Hypothesis test todetermine whether the average annual wage foran advertising manager is different from theaverage annual wage of an auditing manager. Arandom sample of 32 advertising managers fromacross US is taken. The advertising managers arecontacted by telephone and asked what is theirannual salary. A similar random sample is takenof 34 auditing managers. The resulting salarydata are given in the next table, along with thesample means, the sample standard deviationsand the sample variances.
QT-II/Ses 3/SAMPRIT
Crucial points
• The analyst is testing whether there is any difference in the average wage of an advertising manager and an auditing manager
• So it is a two tailed test
• If testing is on whether one was paid more than the other, the test would have been one tailed
QT-II/Ses 3/SAMPRIT
Example
Advertising Managers
74.256 57.791 71.115
96.234 65.145 67.574
89.807 96.767 59.621
93.261 77.242 62.483
103.030 67.056 69.319
74.195 64.276 35.394
75.932 74.194 86.741
80.742 65.360 57.351
39.672 73.904
45.652 54.270
93.083 59.045
63.384 68.508
164.264
253.16
700.70
32
2
1
1
1
1
x
n
411.166
900.12
187.62
34
2
2
2
2
2
x
n
Auditing Managers
69.962 77.136 43.649
55.052 66.035 63.369
57.828 54.335 59.676
63.362 42.494 54.449
37.194 83.849 46.394
99.198 67.160 71.804
61.254 37.386 72.401
73.065 59.505 56.470
48.036 72.790 67.814
60.053 71.351 71.492
66.359 58.653
61.261 63.508
QT-II/Ses 3/SAMPRIT
Working
35.2
34
411.166
32
253.256
0187.62700.70
2
2
2
1
2
1
2121
nS
nS
XXZ
.Hreject not do 1.96, Z 1.96- If
.Hreject 1.96, > or Z 1.96- < ZIf
o
o
.Hreject 1.96, > 2.35 = ZSince o
Rejection
Region
Non Rejection Region
Critical Values
Rejection
Region
cZ 2 33.
025.2
0 cZ 2 33.
025.2
.reject not do ,96.196.1 If
.reject ,96.1or 96.1 If
0
0
Hz
Hzz
35.2
34
411.166
32
253.256
(0)-62.187)-(70.700
()(
2
2
2
1
2
1
)2121
nn
xxz
.reject ,96.135.2 Since 0Hz
96.1cz 96.1cz
QT-II/Ses 3/SAMPRIT
Example
• A sample of 87 working professional working womenshowed that the average amount paid annually into aprivate pension fund per person was $3,352, with asample standard deviation of $1,100. A sample of 76professional working men showed that the averageamount paid annually into a private pension fund perperson was $5,727, with a sample standard deviation of$1,700. A women’s activist group wants to prove thatwomen do not pay as much per year as men into privatepension funds. If they use CI = 99% and the sampledata, will they be able to reject a null hypothesis thatwomen annually pay the same as or more than men intoprivate pension funds?
QT-II/Ses 3/SAMPRIT
Crucial points
• Test is one tailed
• When the population variances areunknown and the sample sizes are large (n1and n2 greater than equal to 30), samplevariances can be used
• For large samples, sample variances aregood approximations of populationvariances
QT-II/Ses 3/SAMPRIT
Working
H
H
o
a
:
:
1 2
1 2
0
0
Non Rejection Region
Critical Value
Rejection
Region
.001
cZ 308. 008.3cz
QT-II/Ses 3/SAMPRIT
Working
Non Rejection Region
Critical Value
Rejection
Region
.001
cZ 308. 0
.H
.H
o
o
reject not do ,08.3 z If
reject 3.08,- <z If
42.10
76
1700
87
1100
057273352
22
2
2
2
1
2
1
2121
nn
xxz
.Horeject 3.08,- < 10.42- = z Since
87
100,1$
352,3$
1
1
1
n
x
Women
76
700,1$
727,5$
2
2
2
n
x
Men
08.3cz
QT-II/Ses 3/SAMPRIT
Confidence Interval to Estimate 1 - 2 When 1, 2
are known
nnzxx
nnzxx
2
2
2
1
2
1
21212
2
2
1
2
1
21
QT-II/Ses 3/SAMPRIT
Example
• A consumer test group wants to determine the difference ingasoline mileage of cars using regular unleaded gas and carsusing premium unleaded gas. Researchers for the group divideda fleet of 100 cars of the same make in half and tested each caron one tank of gas. Fifty of the cars were filled with regularunleaded gas and 50 were filled with premium unleaded gas.The sample average for the regular gasoline group was 21.45miles per gallon, with a standard deviation of 3.46 mpg. Thesample average for the premium gasoline group was 24.6 mpg,with a standard deviation of 2.99 mpg. Construct a 95%confidence interval to estimate the difference in the mean gasmileage between the cars using regular gasoline and the carsusing premium gasoline
QT-II/Ses 3/SAMPRIT
Working
88.142.4
50
99.2 2
50
46.396.16.2445.21
505096.16.2445.21
21
2
21
22
2
2
2
1
2
1
21212
2
2
1
2
1
21
99.246.3
nnxx
nnxx zz
46.3
45.21
50
Re
1
1
1
x
n
gular
99.2
6.24
50
Pr
2
2
2
x
n
emium
1.96 = Confidence %95 z
QT-II/Ses 3/SAMPRIT
The t Test for Differences
in Population Means
• Each of the two populations is normally distributed.
• The two samples are independent.
• The values of the population variances are unknown.
• The variances of the two populations are equal.
12 = 2
2
QT-II/Ses 3/SAMPRIT
t Formula to Test the Difference in
Means Assuming 12 = 2
2
2121
2
2
21
2
1
2121
11
2
)1()1(
)()(
nnnn
nsns
xxt
QT-II/Ses 3/SAMPRIT
Example• At the Hernandez Manufacturing Company, an application of
the test of the difference in small sample mean arises. Newemployees are expected to attend a three-day seminar to learnabout the company. At the end of the seminar, they are tested tomeasure their knowledge about the company. The traditionaltraining method has been lecture and a QnA session.Management decided to experiment with a different trainingprocedure, which processes new employees in two days byusing vcd and having no QnA session. If, this procedure works,it could save the company thousands of dollars over a period ofseveral years. However, there is some concern about theeffectiveness of the two day method, and the companymanagers would like to know whether there is any difference in
the effectiveness of the two training methods.To test the difference in the two methods, the managersrandomly select one group of 15 newly hired employees to takethe three day seminar (method A) and a second group of 12 newemployees for the two day vcd method (method B)
QT-II/Ses 3/SAMPRIT
Example
• The table show the test scores of the two groups. Using α = 0.05,the managers want to determine whether there is a significantdifference in the mean scores of the two groups. They assumethat the scores for this test are normally distributed and that thepopulation variances are approximately equal.
Training Method A
56 51 45
47 52 43
42 53 52
50 42 48
47 44 44
Training Method B
59
52
53
54
57
56
55
64
53
65
53
57
QT-II/Ses 3/SAMPRIT
Solution
H
H
o
a
:
:
1 2
1 2
0
0
If t < - 2.060 or t > 2.060, reject H .
If - 2.060 t 2.060, do not reject H .
o
o
060.2
25212152
025.2
05.
2
25,25.0
21
t
nndf
Rejection
Region
Non Rejection Region
Critical Values
Rejection
Region
2025.
0 . ,.
025 252 060t
2025.
. ,.
025 252 060t
QT-II/Ses 3/SAMPRIT
Solution
Training Method A
56 51 45
47 52 43
42 53 52
50 42 48
47 44 44
Training Method B
59
52
53
54
57
56
55
64
53
65
53
57
495.19
73.47
15
2
1
1
1
s
x
n
273.18
5.56
12
2
2
2
2
s
x
n
QT-II/Ses 3/SAMPRIT
Solution
.Ht oreject -2.060,<-5.20= Since
20.5
12
1
15
1
21215
11273.1814495.19
050.5673.47
11
2
)1()1(
)()(
2121
2
2
21
2
1
2121
nnnn
nsns
xxt
.Ht
.Htt
o
o
reject not do 2.060, 2.060- If
reject 2.060, > or 2.060- < If
QT-II/Ses 3/SAMPRIT
Example• Is there any difference in the way Chinese cultural values affect
the purchasing strategies of industrial buyers in Taiwan andmainland China? A study by researchers at the Nationaluniversity in Taiwan attempted to determine whether there is asignificant difference in the purchasing strategies of industrialbuyers in the two countries based on the cultural dimensionlabeled “integration.” Integration is being in harmony with one’sself, family, and associates. For the study, 46 Taiwanese buyersand 26 mainland Chinese buyers were contacted and interviewed.Buyers were asked to respond to 35 using a 9-point scale withpossible answers ranging from no importance (1) to extremeimportance (9). The resulting statistics for the two groups areshown in the table. Using α = 0.01, test to determine whetherthere is a significant difference between buyers of the twocountries on integration.
QT-II/Ses 3/SAMPRIT
Example
Taiwanese Buyers Mainland Chinese Buyers
Sample size is 46 Sample size is 26
Mean is 5.42 Mean is 5.04
Sample Variance is 0.3364 Sample Variance is 0.2401
QT-II/Ses 3/SAMPRIT
Confidence Interval to Estimate 1 - 2 when
12 and 2
2 are unknown and 12 = 2
2
2 where
11
2
)1()1()(
21
2121
2
2
21
2
121
nndf
nnnn
nsnstxx
QT-II/Ses 3/SAMPRIT
Example
• Construct a 99% Confidence Interval from the following data
Sample Size = 9 Sample Size = 10
Mean = 37.09 Mean = 34.99
Sample sd = 1.727 Sample sd = 1.253
t for 0.005 and 17 is 2.898
QT-II/Ses 3/SAMPRIT
