Transcript Doing Statistics for Business

Doing Statistics for Business
Data, Inference, and Decision Making
Marilyn K. Pelosi
Theresa M. Sandifer
Chapter 10
Comparing Two
Populations
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Chapter 10 Objectives
 Collecting Data From Two Populations
 Large-Sample Test of the Difference in Two
Population Means
 Small-Sample Test of the Difference in Two
Population Means
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Chapter 10 Objectives (con’t)
 Test of the Difference in Two Population
Means -- Paired Data
 Hypothesis Test of the Difference in Two
Population Proportions
 Hypothesis Test of the Difference in Two
Population Variances
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Population 1
Population 2
Sample 1
Sample 2
Figure 10.1 Two Populations and Two Samples
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TRY IT NOW!
The Tissue Manufacturer also recorded values of
MDStrength for 75 tissues made on day 3.
The sample mean MDStrength for that day was found to be
1000.32lb/ream. Is there any evidence that the average MDStrength
is different on day 2 than day 3? Use  = 0.05.
Identify
Population 1:
Population 2:
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TRY IT NOW!
The management of the corporation looking at training
issues are not convinced that they should switch to outside
consultants for all of their training programs. They want to look
at how the two groups compared on the pretest. The relevant data
are given below.
Population
Sample size
Mean score on Pre-test
Standard Deviation of Pretest score
1 (Outside Consultant)
n1 = 50
X 1 = 43.52
s1 = 14.80
2 (In-house Trainer)
n2 = 47
X 2 = 47.66
s2 = 12.80
Test to see if there is a difference in the mean pretest scores. Use  = 0.05.
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Figure 10.2 Two Normally Distributed
Populations with the Same
Variance but Different Means
Populations with the Same Variance but
Different Means
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TRY IT NOW!
The company also wants to look at gender differences in
the test scores for the group of employees trained by the
outside consultant. It assembles the relevant data and find
the following:
Population #
Population description
Sample size
Sample mean
Sample standard deviation
1
Females trained by Outside
Consultant
n1= 27
X 1 = 75.9
2
Males trained by Outside
Consultant
n2= 25
X 2 = 75.2
s1 = 3.9
s2= 4.4
Find the pooled variance for the data.
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(con’t)
Test to see if there is evidence that the mean post-test score
for the females trained by the outside consultant is different
than the mean post-test score for the males trained by the outside
consultant. Use a level of significance of 0.05.
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Figure 10.3 Two Normally Distributed
Populations with Unequal
Variances
Two Normally Distributed Populations with
Unequal Variances
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Pretest and Posttest conditions exist when
data are collected on the same sample
elements before and after some experiment
is performed.
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Dependent Samples are related to each other.
The members of one sample are identical to
or matched or paired with the members in the
other sample according to some characteristic.
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Discovery Exercise 10.1
Introduction to Experimental Design
Did you ever wonder why the restrooms in restaurants have
a sign instructing all employees to wash their hands before returning to
work? The owner of a large restaurant chain wanted to increase the
amount of time that employees washed their hands after reading about
the health implications of washing for only a few seconds. A random
sample of 10 employees was selected. The amount of time each
employee washed was recorded. The data were collected in such a
way that the employees did not know they were being observed.
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Discovery Exercise 10.1
Introduction to Experimental Design
(con’t)
These employees were then educated on the benefits of hand washing.
They watched a health video that detailed the benefits of increasing the
amount of time they washed. One week after the training, employee
and washing was timed again. The times (in seconds) are shown on the
following slide:
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Discovery Exercise 10.1
Introduction to Experimental Design
(con’t)
Employee
1
2
3
4
5
6
7
8
9
10
Before Training
3
3
2
4
3
3
4
3
5
2
After Training
3
3
4
4
5
4
4
4
6
3
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Discovery Exercise 10.1
Introduction to Experimental Design
(con’t)
(a) Use a two-population t test to test the hypothesis that no learning has
occurred. Use a pooled variance.
(b) What is your conclusion?
( c) Why is this surprising (Hint: Of the 10 employees, how many of them
washed at least as long after the training?)
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Discovery Exercise 10.1
Introduction to Experimental Design
(con’t)
(d) Explain how these data are different from most of the two-sample data
sets you have looked at up to this point in the chapter. Why might this
be important?
(e) Suppose 10 different employees were used for the second timing.
Would this make a difference? Explain why or why not.
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Pre and Post Treatment
Post-test
Pre-test
Variable
Observation #
Figure 10.4 Pre- and Post-test Data for
Similar Observations
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Pre and Post Treatments
Posttest
Pre-test
Variable
Observation #
Figure 10.5 Pre- and Post-test Data for
Highly Variable Observations
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TRY IT NOW!
In addition to measuring the number of obstructions
before and after home treatment in the OSAS study,
the number of hypopnea both pre- and post-treatment
for each patient is shown below.
Complete the table.
Hypoponea
Patient # Pre-treatment Post-treatment Differences
1
208
22
2
297
0
3
126
17
4
150
13
5
133
0
6
201
10
7
272
7
8
137
1
9
310
2
d
sD
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E-mail Usage
Test of Two Proportions
The company looking at use of e-mail is also interested in the
use of e-mail for personal messages. After seeing the results
of the last test, it decides that what it really wants to know is if a
higher proportion of women than men use e-mail to send personal
messages. Use a level of significance of 0.05. The data are shown
on the following slide.
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TRY IT NOW!
E-mail Usage
Test of Two Proportions (con’t)
Sample size
Population proportion
Number that use e-mail
Sample proportion
Population 1
Men
35
1
15
Population 2
Women
35
2
18
Finish the table and then complete the hypothesis test..
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F
Two-Sided Test
F
Upper- Tail Test
F
Lower- Tail Test
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Training and Gender
Test for Equality of Variances
When you looked at gender differences for the group of
employees trained by the outside consultant, you did the test using
the pooled variance. That is, you assumed the population variances
were equal. The relevant data were as follows:
Population #
Population description
Sample size
Sample mean
Sample standard deviation
1
Females trained by Outside
Consultant
n1= 27
X 1 = 75.9
2
Males trained by Outside
Consultant
n2= 25
X 2 = 75.2
s1 = 3.9
s2= 4.4
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TRY IT NOW!
Training and Gender
Test for Equality of Variances (con’t)
Assume that the data are normally distributed and perform a test
to see whether or not the assumption of equal variances was reasonable.
Use a level of significance of 0.05.
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Two Populations Hypothesis Tests in Excel
1. Select Tools > Data Analysis menu and from the list
choose zTest: Two Sample for Means.
2. Position cursor in textbox next to Variable 1 Range: and
highlight the range that contains the post-test data for the
first population.
3. Position cursor in the textbox for Variable 2 Range: and
highlight the location of the data for the second
population.
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Two Populations Hypothesis Tests in Excel
(con’t)
4. Position cursor in textbox for Hypothesized Mean
Difference and enter 0.
5. Put cursor in the textbox for Variable 1 Variance (known)
and enter the value of the variance for population 1.
6. Repeat this for Variable 2 Variance (known).
7. If there were labels in any of the data ranges you
highlighted, check Labels checkbox.
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Two Populations Hypothesis Tests in Excel
(con’t)
8. Enter the value of the significance for the test.
9. Finally, indicate where you want the output from the
test to appear.
10. Hit Enter to perform the test.
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Figure 10.7
Dialog Box for Z test for
Two Means
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Figure 10.8
Z test Dialog Box
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Figure 10.9 Output from Z test for
Two Sample Means
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Figure 10.11 Output for t test Assuming
Equal Variances
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Figure 10.12 Output from t test Assuming
Unequal Variances
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Figure 10.13 Output from Paired
Difference Test
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Figure 10.16 Output from Two Sample
Variance Test
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Chapter 10 Summary
In this chapter you have learned:
 About concerns to watch for when collecting data
from two populations
 How to test for the difference in two population
means - large and small samples
 About pre and post data and how to analyze it
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Chapter 10 Summary (cont’d)
 How to test for the difference in two population
proportions
 How to test for the difference in two population
variances
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