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- 1. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-1
Chapter 8
Fundamentals of Hypothesis
Testing: One-Sample Tests
Statistics for Managers
Using Microsoft® Excel
4th Edition
- 2. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-2
Chapter Goals
After completing this chapter, you should be
able to:
Formulate null and alternative hypotheses for
applications involving a single population mean or
proportion
Formulate a decision rule for testing a hypothesis
Know how to use the p-value approaches to test the
null hypothesis for both mean and proportion
problems
Know what Type I and Type II errors are
- 3. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-3
What is a Hypothesis?
A hypothesis is a claim
(assumption) about a
population parameter:
population mean
population proportion
Example: The mean monthly cell phone bill of
this city is μ = $42
Example: The proportion of adults in this city
with cell phones is p = .68
- 4. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-4
States the assumption to be tested
Example: The average number of TV sets in
U.S. Homes is equal to three ( )
Is always about a population parameter,
not about a sample statistic
The Null Hypothesis, H0
3
μ
:
H0
3
μ
:
H0 3
X
:
H0
- 5. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-5
The Null Hypothesis, H0
Begins with the assumption that the null
hypothesis is true
Similar to the notion of innocent until
proven guilty
Refers to the status quo
Always contains “=” , “≤” or “” sign
May or may not be rejected
(continued)
- 6. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-6
The Alternative Hypothesis, H1
Is the opposite of the null hypothesis
e.g.: The average number of TV sets in U.S.
homes is not equal to 3 ( H1: μ ≠ 3 )
Challenges the status quo
Never contains the “=” , “≤” or “” sign
Is generally the hypothesis that is believed (or
needs to be supported) by the researcher
- 7. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-7
Hypothesis Testing
We assume the null hypothesis is true
If the null hypothesis is rejected we have proven
the alternate hypothesis
If the null hypothesis is not rejected we have
proven nothing as the sample size may have
been to small
- 8. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc.
Population
Claim: the
population
mean age is 50.
(Null Hypothesis:
REJECT
Suppose
the sample
mean age
is 20: X = 20
Sample
Null Hypothesis
20 likely if μ = 50?
Is
Hypothesis Testing Process
If not likely,
Now select a
random sample
H0: μ = 50 )
X
- 9. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-9
Do not reject H0 Reject H0
Reject H0
There are two
cutoff values
(critical values),
defining the regions
of rejection
Sampling Distribution of
/2
0
H0: μ = 50
H1: μ 50
/2
Lower
critical
value
Upper
critical
value
50 X
X
20 Likely Sample Results
- 10. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-10
Level of Significance,
Defines the unlikely values of the sample statistic if
the null hypothesis is true
Defines rejection region of the sampling distribution
Is designated by , (level of significance)
Typical values are .01, .05, or .10
Is the compliment of the confidence coefficient
Is selected by the researcher before sampling
Provides the critical value of the test
- 11. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-11
Level of Significance
and the Rejection Region
H0: μ ≥ 3
H1: μ < 3
0
H0: μ ≤ 3
H1: μ > 3
Represents
critical value
Lower tail test
Level of significance =
0
Upper tail test
Two tailed test
Rejection
region is
shaded
/2
0
/2
H0: μ = 3
H1: μ ≠ 3
- 12. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-12
Type I Error
When a true null hypothesis is rejected
The probability of a Type I Error is
Called level of significance of the test
Set by researcher in advance
Type II Error
Failure to reject a false null hypothesis
The probability of a Type II Error is β
Errors in Making Decisions
- 13. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-13
Example
The Truth
Verdict
Innocent No
error
Type II Error
Guilty Type I Error
Possible Jury Trial Outcomes
Guilty
Innocent
No Error
- 14. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-14
Outcomes and Probabilities
Actual
Situation
Decision
Do Not
Reject
H0
No error
(1 - )
Type II Error
( β )
Reject
H0
Type I Error
( )
Possible Hypothesis Test Outcomes
H0 False
H0 True
Key:
Outcome
(Probability)
No Error
( 1 - β )
- 15. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-15
Type I & II Error Relationship
Type I and Type II errors can not happen at
the same time
Type I error can only occur if H0 is true
Type II error can only occur if H0 is false
If Type I error probability ( ) , then
Type II error probability ( β )
- 16. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-16
p-Value Approach to Testing
p-value: Probability of obtaining a test
statistic more extreme ( ≤ or ) than the
observed sample value given H0 is true
Also called observed level of significance
- 17. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-17
p-Value Approach to Testing
Convert Sample Statistic (e.g. ) to Test
Statistic (e.g. t statistic )
Obtain the p-value from a table or computer
Compare the p-value with
If p-value < , reject H0
If p-value , do not reject H0
X
(continued)
- 18. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-18
9 Steps in
Hypothesis Testing
1. State the null hypothesis, H0
2. State the alternative hypotheses, H1
3. Choose the level of significance, α
4. Choose the sample size, n
5. Determine the appropriate test statistic to use
6. Collect the data
7 Compute the p-value for the test statistic from the
sample result
8. Make the statistical decision: Reject H0 if the p-value
is less than alpha
9. Express the conclusion in the context of the problem
- 19. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-19
Hypothesis Tests for the Mean
Known Unknown
Hypothesis
Tests for
- 20. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-20
Hypothesis Testing Example
Test the claim that the true mean # of TV
sets in U.S. homes is equal to 3.
1-2. State the appropriate null and alternative
hypotheses
H0: μ = 3 H1: μ ≠ 3 (This is a two tailed test)
3. Specify the desired level of significance
Suppose that = .05 is chosen for this test
4. Choose a sample size
Suppose a sample of size n = 100 is selected
- 21. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-21
5. Determine the appropriate Test
σ is unknown so this is a t test
6. Collect the data
Suppose the sample results are
n = 100, = 2.84 s = 0.8
7. So the test statistic is:
The p value for n=100, =.05, t=-2 is .048
2.0
.08
.16
100
0.8
3
2.84
n
s
μ
X
t
Hypothesis Testing Example
(continued)
X
- 22. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-22
Reject H0 Do not reject H0
8. Is the test statistic in the rejection region?
= .05/2
-t= -1.98 0
Reject H0 if p
is < alpha;
otherwise do
not reject H0
Hypothesis Testing Example
(continued)
= .05/2
Reject H0
+t= +1.98
Here, t = -2.0 < -1.98, so the test
statistic is in the rejection region
The p-value .048 is < alpha .05, we
reject the null hypothesis
- 23. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-23
9. Express the conclusion in the context of the problem
Since The p-value .048 is < alpha .05,
we have rejected the null hypothesis
Thereby proving the alternate hypothesis
Conclusion: There is sufficient evidence that the mean
number of TVs in U.S. homes is not equal to 3
Hypothesis Testing Example
(continued)
If we had failed to reject the null hypothesis the
conclusion would have been: There is not sufficient
evidence to reject the claim that the mean number of
TVs in U.S. home is 3
- 24. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-24
One Tail Tests
In many cases, the alternative hypothesis
focuses on a particular direction
H0: μ ≥ 3
H1: μ < 3
H0: μ ≤ 3
H1: μ > 3
This is a lower tail test since the
alternative hypothesis is focused on
the lower tail below the mean of 3
This is an upper tail test since the
alternative hypothesis is focused on
the upper tail above the mean of 3
- 25. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-25
Reject H0 Do not reject H0
There is only one
critical value, since
the rejection area is
in only one tail
Lower Tail Tests
-t 3
H0: μ ≥ 3
H1: μ < 3
Critical value
X
- 26. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-26
Reject H0
Do not reject H0
Upper Tail Tests
tα
3
H0: μ ≤ 3
H1: μ > 3
There is only one
critical value, since
the rejection area is
in only one tail
Critical value
t
X
- 27. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-27
Assumptions of the One-Sample t Test
The data is randomly selected
The population is normally distributed or
the sample size is over 30 and the population is
not highly skewed
- 28. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-28
Hypothesis Tests for Proportions
Involves categorical values
Two possible outcomes
“Success” (possesses a certain characteristic)
“Failure” (does not possesses that characteristic)
Fraction or proportion of the population in the
“success” category is denoted by p
- 29. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-29
Proportions
Sample proportion in the success category is
denoted by ps
When both np and n(1-p) are at least 5, ps
can be approximated by a normal distribution
with mean and standard deviation
size
sample
sample
in
successes
of
number
n
X
ps
p
μ s
p
n
p)
p(1
σ s
p
(continued)
- 30. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-30
The sampling
distribution of ps
is approximately
normal, so the test
statistic is a Z
value:
Hypothesis Tests for Proportions
n
)
p
(
p
p
p
Z
s
1
np 5
and
n(1-p) 5
Hypothesis
Tests for p
np < 5
or
n(1-p) < 5
Not discussed
in this chapter
- 31. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-31
An equivalent form
to the last slide,
but in terms of the
number of
successes, X:
Z Test for Proportion
in Terms of Number of Successes
)
p
1
(
np
np
X
Z
X 5
and
n-X 5
Hypothesis
Tests for X
X < 5
or
n-X < 5
Not discussed
in this chapter
- 32. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-32
Example: Z Test for Proportion
A marketing company
claims that it receives
8% responses from its
mailing. To test this
claim, a random sample
of 500 were surveyed
with 25 responses. Test
at the = .05
significance level.
Check:
np = (500)(.08) = 40
n(1-p) = (500)(.92) = 460
- 33. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-33
Z Test for Proportion: Solution
= .05
n = 500, ps = .05
p-value for -2.27 is .0134
Decision:
Reject H0 at = .05
H0: p = .08
H1: p .08
Critical Values: ± 1.96
Test Statistic:
Conclusion:
z
0
Reject Reject
.025
.025
1.96
-2.47
There is sufficient
evidence to reject the
company’s claim of 8%
response rate.
2.47
500
.08)
.08(1
.08
.05
n
p)
p(1
p
p
Z
s
-1.96
- 35. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-35
Sample PHStat Output
Input
Output
- 36. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-36
Potential Pitfalls and
Ethical Considerations
Use randomly collected data to reduce selection biases
Do not use human subjects without informed consent
Choose the level of significance, α, before data
collection
Do not employ “data snooping” to choose between one-
tail and two-tail test, or to determine the level of
significance
Do not practice “data cleansing” to hide observations
that do not support a stated hypothesis
Report all pertinent findings
- 37. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-37
Chapter Summary
Addressed hypothesis testing methodology
Discussed critical value and p–value approaches to
hypothesis testing
Discussed type 1 and Type2 errors
Performed two tailed t test for the mean (σ unknown)
Performed Z test for the proportion
Discussed one-tail and two-tail tests
Addressed pitfalls and ethical issues
- 38. Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 8-38
Answer Sheet for All Problems
___________ Null Hypothesis
___________ Alternate Hypothesis
___________ Alpha
___________ p-value
___________ Decision (reject or do not reject)
Conclusion: