Transcript Basic Business Statistics (9th Edition)

Basic Business Statistics
(9th Edition)
Chapter 9
Fundamentals of Hypothesis
Testing: One-Sample Tests
© 2004 Prentice-Hall, Inc.
Chap 9-1
Chapter Topics
Hypothesis Testing Methodology
 Z Test for the Mean ( Known)
 p-Value Approach to Hypothesis Testing
 Connection to Confidence Interval Estimation
 One-Tail Tests
 t Test for the Mean (  Unknown)
 Z Test for the Proportion
2
  Test for the Variance or Standard Deviation
 Potential Hypothesis-Testing Pitfalls and
Ethical Issues

© 2004 Prentice-Hall, Inc.
Chap 9-2
What is a Hypothesis?

A Hypothesis is a
Claim (Assertion)
about the Population
Parameter


I claim the mean GPA of
this class is   3.5!
Examples of parameters
are population mean (μ)
or proportion (P )
The parameter must
be identified before
analysis
© 1984-1994 T/Maker Co.
© 2004 Prentice-Hall, Inc.
Chap 9-3
The Null Hypothesis, H0

States the Claim or Assertion to be Tested




E.g., The mean GPA is 3.5
H 0 :   3.5
Null Hypothesis is Always about a Population
Parameter ( H 0 :   3.5), Not about a Sample
Statistic ( H0 : X  3.5 )
Is the Hypothesis a Researcher Tries to Reject
© 2004 Prentice-Hall, Inc.
Chap 9-4
The Null Hypothesis, H0

Begin with the Assumption that the Null
Hypothesis is True




(continued)
Similar to the notion of innocent until
proven guilty
Refers to the Status Quo
Always Contains the “=” Sign
The Null Hypothesis May or May Not be
Rejected
© 2004 Prentice-Hall, Inc.
Chap 9-5
The Alternative Hypothesis, H1

Is the Opposite of the Null Hypothesis




E.g., The mean GPA is NOT 3.5 ( H1
:   3.5 )
Challenges the Status Quo
Never Contains the “=” Sign
Is Generally the Hypothesis that the
Researcher is Interested in
© 2004 Prentice-Hall, Inc.
Chap 9-6
Error in Making Decisions

Type I Error

Reject a true null hypothesis



When the null hypothesis is rejected, we can
say that “We have shown the null hypothesis
to be false (with some ‘slight’ probability, i.e.
 , of making a wrong decision)
Has serious consequences
Probability of Type I Error is 
Called level of significance
 Set by researcher

© 2004 Prentice-Hall, Inc.
Chap 9-7
Error in Making Decisions

Type II Error



Fail to reject a false null hypothesis
Probability of making a Type II Error is

The Probability of Not Making a Type II Error
1   



(continued)
Called the Power of the Test
Probability of Not Making Type I Error


1   
Called the Confidence Coefficient
© 2004 Prentice-Hall, Inc.
Chap 9-8
Hypothesis Testing Process
Assume the
population
mean GPA is 3.5
(H 0 :   3.5)
Identify the Population
Is X  2.4 likely if   3.5?
No, not likely!
REJECT
Null Hypothesis
© 2004 Prentice-Hall, Inc.
Take a Sample
 X  2.4
Chap 9-9
Reason for Rejecting H0
Sampling Distribution of
X
... Therefore,
we reject the
null hypothesis
that  = 3.5.
It is unlikely that
we would get a
sample mean of
this value ...
... if in fact this were
the population mean.
2.4
© 2004 Prentice-Hall, Inc.
 = 3.5
If H0 is true
X
Chap 9-10
Level of Significance, 

Defines Unlikely Values of Sample Statistic if
Null Hypothesis is True


Designated by




Called rejection region of the sampling distribution

, (level of significance)
Typical values are .01, .05, .10
Selected by the Researcher at the Beginning
Controls the Probability of Committing a Type
I Error
Provides the Critical Value(s) of the Test
© 2004 Prentice-Hall, Inc.
Chap 9-11
Level of Significance and the
Rejection Region

H0:  3.5
H1:  < 3.5
H0:   3.5
H1:  > 3.5
Rejection
Regions
0
0
H0:  3.5
H1:   3.5
Critical
Value(s)

/2
0
© 2004 Prentice-Hall, Inc.
Chap 9-12
Result Probabilities
H0: Innocent
Jury Trial
Hypothesis Test
The Truth
Guilty Decision
Verdict Innocent
Correct
Innocent
Decision
Guilty
© 2004 Prentice-Hall, Inc.
Type I
Error
Type II
Error
Do Not
Reject
H0
Correct
Decision
Reject
H0
The Truth
H 0True
H0 False
Confidence
Type II
Error   
1   
Type I
Error
Power
  1   
Chap 9-13
Type I & II Errors Have an
Inverse Relationship
Reduce probability of one error
and the other one goes up holding
everything else unchanged.


© 2004 Prentice-Hall, Inc.
Chap 9-14
Factors Affecting Type II Error

True Value of Population Parameter



increases when

increases when


decreases

Population Standard Deviation


increases when the difference between the
hypothesized parameter and its true value
decreases
Significance Level




increases
Sample Size


© 2004 Prentice-Hall, Inc.
 
increases when n decreases

n
Chap 9-15
How to Choose between Type I
and Type II Errors


Choice Depends on the Cost of the Errors
Choose Smaller Type I Error When the Cost of
Rejecting the Maintained Hypothesis is High



A criminal trial: convicting an innocent person
The Exxon Valdez: causing an oil tanker to sink
Choose Larger Type I Error When You Have
an Interest in Changing the Status Quo


A decision in a startup company about a new piece
of software
A decision about unequal pay for a covered group
© 2004 Prentice-Hall, Inc.
Chap 9-16
Critical Values Approach to
Testing


Convert Sample Statistic (e.g., X ) to
Test Statistic (e.g., Z, or t statistic)
Obtain Critical Value(s) for a Specified 
from a Table or Computer


If the test statistic falls in the critical region,
reject H0
Otherwise, do not reject H0
© 2004 Prentice-Hall, Inc.
Chap 9-17
p-Value Approach to Testing


Convert Sample Statistic (e.g., X ) to Test
Statistic (e.g., Z, or t statistic)
Obtain the p-value from a table or computer




p-value: probability of obtaining a test statistic as
extreme or more extreme (  or  ) than the
observed sample value given H0 is true
Called observed level of significance
Smallest value of  that an H0 can be rejected
Compare the p-value with 


© 2004 Prentice-Hall, Inc.
If p-value
If p-value
 
 
, do not reject H0
, reject H0
Chap 9-18
General Steps in Hypothesis
Testing
E.g., Test the Assumption that the True Mean # of
TV Sets in U.S. Homes is at Least 3 (  Known)
1. State the H0
H0 :   3
2. State the H1
H1 :   3
3. Choose
 =.05

4. Choose n
5. Choose Test
© 2004 Prentice-Hall, Inc.
n  100
Z test
Chap 9-19
General Steps in Hypothesis
Testing
(continued)
6. Set up critical value(s)
Reject H0

Z
-1.645
7. Collect data
100 households surveyed
8. Compute test statistic
and p-value
Computed Z =-2,
p-value = .0228
9. Make statistical decision Reject null hypothesis
10.Express conclusion
© 2004 Prentice-Hall, Inc.
The true mean # TV set is
less than 3
Chap 9-20
One-Tail Z Test for Mean
(  Known)

Assumptions





Population is normally distributed
If not normal, requires large samples
Null hypothesis has  or  sign only
 is known
Z Test Statistic

© 2004 Prentice-Hall, Inc.
X 
Z
/ n
Chap 9-21
Rejection Region
H0: 0
H1:  > 0
H0: 0
H1:  < 0
Reject H0
Reject H0


0
Z must be significantly
below 0 to reject H0
© 2004 Prentice-Hall, Inc.
Z
0
Z
Small values of Z don’t
contradict H0 ; don’t
reject H0 !
Chap 9-22
Example: One-Tail Test
Does an average box of
cereal contain more than
368 grams of cereal? A
random sample of 25 boxes
showed X = 372.5. The
company has specified  to
be 15 grams. Test at the 
0.05 level.
© 2004 Prentice-Hall, Inc.
368 gm.
H0: 368
H1: > 368
Chap 9-23
Reject and Do Not Reject
Regions
H0 :   368
Reject
.05
Do Not Reject
X
 X  368
372.5
0
1.645
Z
1.5
0
© 2004 Prentice-Hall, Inc.
H1 :   368
Chap 9-24
Finding Critical Value: One-Tail
What is Z given  = 0.05?
Standardized Cumulative
Normal Distribution Table
(Portion)
Z 1
Z
.95
 = .05
0 1.645 Z
Critical Value
= 1.645
© 2004 Prentice-Hall, Inc.
.04
.05
.06
1.6 .9495 .9505 .9515
1.7 .9591 .9599 .9608
1.8 .9671 .9678 .9686
1.9 .9738 .9744 .9750
Chap 9-25
Example Solution: One-Tail Test
H0: 368
H1:  > 368
Test Statistic:
 = 0.5
n = 25
Critical Value: 1.645
Reject
.05
0 1.645 Z
© 2004 Prentice-Hall, Inc.
1.50
X 
Z
 1.50

n
Decision:
Do Not Reject at  = .05.
Conclusion:
Insufficient Evidence that
True Mean is More Than 368.
Chap 9-26
p -Value Solution
p-Value is P(Z 1.50) = 0.0668
Use the
alternative
hypothesis
to find the
direction of
the rejection
region.
© 2004 Prentice-Hall, Inc.
p-Value =.0668
1.0000
- .9332
.0668
0
1.50
From Z Table:
Lookup 1.50 to
Obtain .9332
Z
Z Value of Sample
Statistic
Chap 9-27
p -Value Solution
(continued)
(p-Value = 0.0668)  ( = 0.05)
Do Not Reject.
p Value = 0.0668
Reject
 = 0.05
0
1.50
1.645
Z
Test Statistic 1.50 is in the Do Not Reject Region
© 2004 Prentice-Hall, Inc.
Chap 9-28
One-Tail Z Test for Mean
(  Known) in PHStat


PHStat | One-Sample Tests | Z Test for the
Mean, Sigma Known …
Example in Excel Spreadsheet
© 2004 Prentice-Hall, Inc.
Chap 9-29
Example: Two-Tail Test
Does an average box of
cereal contain 368 grams of
cereal? A random sample
of 25 boxes showed X =
372.5. The company has
specified  to be 15 grams
and the distribution to be
normal. Test at the 
0.05 level.
© 2004 Prentice-Hall, Inc.
368 gm.
H0:  368
H1:   368
Chap 9-30
Reject and Do Not Reject
Regions
H0 :   368
Reject
Reject
.025
.025
X
 X  368
-1.96
372.5
0
1.96
Z
1.5
0
© 2004 Prentice-Hall, Inc.
H1 :   368
Chap 9-31
Example Solution: Two-Tail Test
H0: 368
H1:  368
Test Statistic:
X   372.5  368
Z

 1.50

15
n
25
 = 0.05
n = 25
Critical Value: ±1.96
Reject
.025
-1.96
© 2004 Prentice-Hall, Inc.
.025
0 1.96
1.50
Z
Decision:
Do Not Reject at  = .05.
Conclusion:
Insufficient Evidence that
True Mean is Not 368.
Chap 9-32
p-Value Solution
(p-Value = 0.1336)  ( = 0.05)
Do Not Reject.
p-Value = 2 x 0.0668
Reject
Reject
 = 0.05
0
1.50
1.96
Z
Test Statistic 1.50 is in the Do Not Reject Region
© 2004 Prentice-Hall, Inc.
Chap 9-33
Two-Tail Z Test for Mean
(  Known) in PHStat


PHStat | One-Sample Tests | Z Test for the
Mean, Sigma Known …
Example in Excel Spreadsheet
© 2004 Prentice-Hall, Inc.
Chap 9-34
Connection to Confidence
Intervals
For X  372.5,   15 and n  25,
the 95% confidence interval is:
372.5  1.96  15 / 25    372.5  1.96  15 / 25
or
366.62    378.38
We are 95% confident that the population mean is
between 366.62 and 378.38.
If this interval contains the hypothesized mean (368),
we do not reject the null hypothesis.
© 2004 Prentice-Hall, Inc.
It does. Do not reject.
Chap 9-35
t Test:  Unknown

Assumption




Population is normally distributed
If not normal, requires a large sample
 is unknown
t Test Statistic with n-1 Degrees of Freedom

© 2004 Prentice-Hall, Inc.
X 
t
S/ n
Chap 9-36
Example: One-Tail t Test
Does an average box of
cereal contain more than
368 grams of cereal? A
random sample of 36
boxes showed X = 372.5,
and s  15. Test at the 
0.01 level.
 is not given
© 2004 Prentice-Hall, Inc.
368 gm.
H0:  368
H1:  368
Chap 9-37
Reject and Do Not Reject
Regions
H0 :   368
Reject
.01
Do Not Reject
X
 X  368
372.5
0
2.4377
t35
1.8
0
© 2004 Prentice-Hall, Inc.
H1 :   368
Chap 9-38
Example Solution: One-Tail
H0: 368
H1:  368
Test Statistic:
 = 0.01
n = 36, df = 35
Critical Value: 2.4377
Reject
.01
0 2.437
© 2004 Prentice-Hall, Inc.
7
1.80
t35
X   372.5  368
t

 1.80
S
15
n
36
Decision:
Do Not Reject at a = .01.
Conclusion:
Insufficient Evidence that
True Mean is More Than 368.
Chap 9-39
p -Value Solution
(p-Value is between .025 and .05)  ( = 0.01)
Do Not Reject.
p-Value = [.025, .05]
Reject
 = 0.01
0
t35
2.4377
Test Statistic 1.80 is in the Do Not Reject Region
© 2004 Prentice-Hall, Inc.
1.80
Chap 9-40
t Test:  Unknown in PHStat


PHStat | One-Sample Tests | t Test for the
Mean, Sigma Known …
Example in Excel Spreadsheet
© 2004 Prentice-Hall, Inc.
Chap 9-41
Proportion


Involves Categorical Variables
Two Possible Outcomes


“Success” (possesses a certain characteristic) and
“Failure” (does not possess a certain characteristic)
Fraction or Proportion of Population in the
“Success” Category is Denoted by p
© 2004 Prentice-Hall, Inc.
Chap 9-42
Proportion

Sample Proportion in the Success Category is
Denoted by pS


(continued)
X Number of Successes
ps  
n
Sample Size
When Both np and n(1-p) are at Least 5, pS
Can Be Approximated by a Normal Distribution
with Mean and Standard Deviation

p  p
s
© 2004 Prentice-Hall, Inc.
p(1  p)
 ps 
n
Chap 9-43
Example: Z Test for Proportion
A marketing company
claims that a survey
will have a 4% response
rate. To test this claim,
a random sample of 500
were surveyed with 25
responses. Test at the 
= .05 significance level.
© 2004 Prentice-Hall, Inc.
Check:
np  500 .04   20
5
n 1  p   500 1  .04 
 480  5
Chap 9-44
Reject and Do Not Reject
Regions
H0 : p  0.04
Reject
Reject
.025
.025
P  p  0.04 0.05
PS
S
-1.96
0
1.96
Z
1.1411
© 2004 Prentice-Hall, Inc.
H1 : p  0.04
Chap 9-45
Z Test for Proportion: Solution
Test Statistic:
H0: p .04
H1: p  .04
Z
 = .05
n = 500
Critical Values:  1.96
Reject
Reject
.025
0.05
0.04
-1.96 0 1.96
© 2004 Prentice-Hall, Inc.
1.1411
pS  p
p 1  p 
n

.05  .04
.04 1  .04 
500
 1.1411
Decision:
Do not reject at  = .05.
.025
PS
Z
Conclusion:
We do not have sufficient
evidence to reject the
company’s claim of 4%
response rate.
Chap 9-46
p -Value Solution
(p-Value = 0.2538)  ( = 0.05)
Do Not Reject.
p-Value = 2 x .1269
Reject
Reject
 = 0.05
0
1.1411
1.96
Z
Test Statistic 1.1411 is in the Do Not Reject Region
© 2004 Prentice-Hall, Inc.
Chap 9-47
Z Test for Proportion in PHStat


PHStat | One-Sample Tests | Z Test for the
Proportion …
Example in Excel Spreadsheet
© 2004 Prentice-Hall, Inc.
Chap 9-48
 Test for Variance
2
or Standard Deviation

Assumption


Population is normally distributed
Test Statistic

 
2
2
n

1
S
 
2
where n  sample size
S  sample variance
2
 2  hypothesized population variance
© 2004 Prentice-Hall, Inc.
Chap 9-49
Example:  Test for
Standard Deviation
2
Has the standard deviation
of the weight of cereal
boxes produced by a
production process
changed from the specified
level of 15 grams? A
sample of 25 cereal boxes
shows a sample standard
deviation of 17.7 grams.
Test at a 5% level of
significance.
© 2004 Prentice-Hall, Inc.
368 gm.
H 0 :   15 grams
( 2  225 grams squared)
H1 :   15 grams
( 2  225 grams squared)
Chap 9-50
 Test for Standard Deviation:
2
Solution
Test Statistic:
H 0 :   15 grams
H1 :   15 grams
  0.05
n  25
Critical Values:
12.401 and 39.364
Reject
0.025
12.401
© 2004 Prentice-Hall, Inc.
Reject
0.95
33.42
0.025
39.364
n  1 S

 
2
2

2
25  117.7 


2
15
2
 33.42
Decision:
Do not reject at   0.05
Conclusion:
There is insufficient evidence
that the standard deviation of
the process has changed from
the specified level of 15
Chap 9-51
grams
 Test for Variance or
2
Standard Deviation in PHStat


PHStat | One-Sample Test | Chi-Square Test
for Variance …
Example in Excel Spreadsheet
© 2004 Prentice-Hall, Inc.
Chap 9-52
Potential Pitfalls and
Ethical Issues



Data Collection Method is Not Randomized to
Reduce Selection Biases
Treatment of Human Subjects are Manipulated
Without Informed Consent
Data Snooping is Used to Choose between
One-Tail and Two-Tail Tests, and to Determine
the Level of Significance
© 2004 Prentice-Hall, Inc.
Chap 9-53
Potential Pitfalls and
Ethical Issues


(continued)
Data Cleansing is Practiced to Hide
Observations that do not Support a Stated
Hypothesis
Fail to Report Pertinent Findings
© 2004 Prentice-Hall, Inc.
Chap 9-54
Chapter Summary

Addressed Hypothesis Testing Methodology

Performed Z Test for the Mean (  Known)

Discussed p –Value Approach to Hypothesis
Testing

Made Connection to Confidence Interval
Estimation
© 2004 Prentice-Hall, Inc.
Chap 9-55
Chapter Summary
(continued)

Performed One-Tail and Two-Tail Tests

Performed t Test for the Mean (  Unknown)

Performed Z Test for the Proportion

Performed

2
Test for Variance or Standard
Deviation

Discussed Potential Pitfalls and Ethical Issues
© 2004 Prentice-Hall, Inc.
Chap 9-56