INFERENCE: TWO POPULATIONS

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Business Statistics (BUSA 3101)Dr. Lari H.
Arjomandlariarjomand_at_clayton.edu
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Chapter 11 Hypothesis Testing About the
Differences Between Two Population Means

• H0
• µ1 - µ2 0

• Ha
• µ1 - µ2 0

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Hypothesis Testing About the Differences Between
Two Population Means

• Let ?1 equal the mean of population 1 and ?2
  equal
• the mean of population 2.

• The difference between the two population means
  is
• ?1 - ?2.

• Let n1 be the sample size of population 1 and n2
  the sample size of population 2.

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Hypothesis Tests About m 1 - m 2s 1 and s 2
Known

• Hypotheses

Left-tailed
Right-tailed
Two-tailed

• Test Statistic (Actual z Value)

Where D0 can be any number
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Challenging Example 1
Two cities, Bradford and Kane are separated only
by the Conewango River. There is competition
between the two cities. The local

paper recently reported that the mean household
income in Bradford is 38,000 with a standard
deviation of 6,000 for a sample of 40
households. The same article reported the mean
income in Kane is 35,000
with a standard deviation of 7,000 for a sample
of 35 households. At the 0.01 significance level
can we conclude the mean income in Bradford is
more?
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Solution
Step 3 Find the appropriate test statistic.
Because both samples are more than 30, we can use
z as the test statistic.
Step 4 State the decision rule. The null
hypothesis is rejected if actual z is greater
than critical z of 2.33 or if p .01
Step 2 State the level of significance. The
0.01 significance level is stated in the problem.
Step 1 State the null and alternate
hypotheses. H0 µB lt µK H1 µB gt µK
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Step 5 Compute the value of actual z and make a
decision
Actual Z
Because the actual Z 1.98 lt critical Z 2.33,
and since the p-value 0.0239 gt a 0.01 the
decision is to accept the null hypothesis. Thus
we cannot conclude that the mean household income
in Bradford is larger then the mean household
income in Kane.
H0 µB lt µK H1 µB gt µK
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Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown and Small Sample
When s 1 and s 2 are unknown, we will

• use the sample standard deviations s1 and s2
• as estimates of s 1 and s 2 , and

• use t table instead of Z table.
• Assuming at least one of the sample is n lt 30

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Hypothesis Tests About m 1 - m 2s 1 and s 2
Unknown and Small Sample

• Hypotheses

Left-tailed
Right-tailed
Two-tailed

• Test Statistic (Actual t)When n lt 30 and s is
  unknown

Where D0 can be any number
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EXCEL APPLICATION
When in a problem raw data are given, then you
may use either Data Analysis Add-Ins or SWStat
Add-Ins in Excel to solve the problem. See next
slides.
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Hypothesis Tests About m 1 m 2 Whens 1 and s 2
Unknown Small SamplesChallenging Example
1Using SWStat and Data Analysis in Excel

• Ariana Corporation wants to increase the
  productivity of its line workers. To do so, two
  different programs have been suggested to help
  increase productivity. Twenty employees, making
  up a sample, have been randomly assigned to one
  of the two programs and their output for a day's
  work has been recorded. The results are given on
  the next slide.

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Example 1 Continued
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Example 1 ContinuedQuestions

• a. State the null and alternative hypotheses.
• b. Use Excel, and at 95 confidence test to
  determine if the means of the two
  populations are equal, and
• c. Explain your answer

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Hypothesis Tests About m 1 - m 2 Whens 1 and s 2
Unknown Small SamplesUsing Data Analysis in
Excel

• Excels t-Test Two Sample Assuming Unequal
• Variances Tool

Step 1 Select the Tools menu
Step 2 Choose the Data Analysis option
Step 3 Choose t-Test Two Sample Assuming
Unequal Variances from the list of
Analysis Tools
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Solution to 1 Using Data Analysis in Excel

• Excel Dialog Box

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Solution to 1 Using Data Analysis in Excel
H0 ?1 ?2 Ha ?1 ?2
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Hypothesis Tests About m 1 - m 2 Whens 1 and s 2
Unknown Small Samples

• SWStat t-Test Two Sample Assuming Unequal
  Variances Tool

Step 1 Create Data area
Step 2 Choose Statistics, and then choose
the Intervals and Tests option
Step 3 From Type, Choose Two Samples,
Different Means, Unequal Variances, t-Test
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Solution to 1 Using SWStat(Creating Data Area)
Data Area
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Solution to 1 Using SWStat(Selecting Intervals
and Tests Option)
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Solution to 1 Using SWStat(Results)
H0 ?1 ?2 Ha ?1 ?2
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Hypothesis Tests About m 1 m 2 Whens 1 and s 2
UnknownChallenging Example 2

• Information regarding the ACT scores of samples
  of twenty two students in two different majors at
  CSU is given in next slide.
• Questions (a) Use Excel, and at 95 confidence
  test to determine whether there is a significant
  difference in the means of the two populations,
  and (b) explain your answer.

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DATA
Hypothesis Test
H0 ?1 ?2 Ha ?1 ?2
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Solution to 2 Using Data Analysis in Excel
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Solution to 2 Using SWStat(Results)
Two Samples, Different Means, Unequal Variances,
t Test
H0 ?1 ?2 Ha ?1 ?2
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Hypothesis Tests About m 1 - m 2 Whens 1 and s 2
are Known

• If samples are large (n130 and n230) and also
  if population standard deviations (s1 and s2) are
  given then you can use either Data Analysis or
  SWStat in Excel to solve the problem.
• Note that in this case, the option that you use
  in Data Analysis is z-Test Two Sample for Means
  and the option that you use in SWStat is Two
  Sample, diff Variances/SDs F
• See next example.

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Hypothesis Tests About m 1 - m 2 Whens 1 and s 2
are Known

• Example Par, Inc.

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.
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Hypothesis Testing of ?1 - ?2 s 1 and s 2 Known

• Example Par, Inc.

Sample 1 Par, Inc.
Sample 2 Rap, Ltd.
Sample Size
120 balls 80 balls
Sample Mean
275 yards 258 yards
Given that the two population standard deviations
are known as s 1 15 yards and s 2 20 yards,
then
Can we conclude, using a 0.01, that the mean
driving distance of Par, Inc. golf balls is
greater than the mean driving distance of Rap,
Ltd. golf balls?
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Hypothesis Testing of ?1 - ?2 When s 1 and s 2
are Known

• p Value and Critical Value Approaches

The hypotheses is
H0 ?1 - ?2 lt 0 ? Ha ?1 - ?2 gt 0

• where
• ?1 mean distance for the population
• of Par, Inc. golf balls
• ?2 mean distance for the population
• of Rap, Ltd. golf balls

The level of significance is
a .01
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Hypothesis Tests About m 1 - m 2 Whens 1 and s 2
are Known

• Excels z-Test Two Sample for Means Tool

Step 1 After data entry, then select the Tools
menu
Step 2 Choose the Data Analysis option
Step 3 Choose z-Test Two Sample for Means
from the list of Analysis Tools
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Hypothesis Tests About m 1 - m 2 When s 1 and s
2 are Known

• Excel Dialog Box

H0 ?1 - ?2 lt 0 ? Ha ?1 - ?2 gt 0
H0 ?1 ?2 ? Ha ?1 gt ?2
OR
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Hypothesis Tests About m 1 - m 2 Whens 1 and s 2
are Known

• Excel Value Worksheet

Note Rows 14-121 are not shown.
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Conclusion

• Because pvalue 0 lt a 0.01 and also because
  actual z 6.49 gt critical z 2.33 we reject H0.
  In other words

• At the 0.01 level of significance, the sample
  evidence indicates that the mean driving distance
  of Par, Inc. golf balls ( µ1) is greater than the
  mean driving distance of Rap, Ltd. golf balls
  (µ2). In other words

H0 ?1 - ?2 lt 0 ? Ha ?1 - ?2 gt 0
H0 ?1 ?2 ? Ha ?1 gt ?2
OR
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End of Chapter 11
DANCER