Basic Business Statistics (8th Edition)
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Transcript Basic Business Statistics (8th Edition)
Statistics for Managers
using Excel
3rd Edition
Chapter 7
Fundamentals of Hypothesis
Testing: One-Sample Tests
© 2002 Prentice-Hall, Inc.
Chap 7-1
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
Potential hypothesis-testing pitfalls and ethical
considerations
© 2002 Prentice-Hall, Inc.
Chap 7-2
What is a Hypothesis?
A hypothesis is a
claim (assumption)
about the population
parameter
I claim the mean GPA of
this class is 3.5!
Examples of parameters
are population mean
or proportion
The parameter must
be identified before
analysis
© 1984-1994 T/Maker Co.
© 2002 Prentice-Hall, Inc.
Chap 7-3
The Null Hypothesis, H0
States the assumption (numerical) to be
tested
e.g.: The average number of TV sets in U.S.
Homes is at least three (H 0 : 3 )
Is always about a population parameter
( H0 : 3), not about a sample
statistic ( H0 : X 3 )
© 2002 Prentice-Hall, Inc.
Chap 7-4
The Null Hypothesis, H0
Begins 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
May or may not be rejected
© 2002 Prentice-Hall, Inc.
Chap 7-5
The Alternative Hypothesis, H1
Is the opposite of the null hypothesis
e.g.: The average number of TV sets in U.S.
homes is less than 3 ( H1 : 3 )
Challenges the status quo
Never contains the “=” sign
May or may not be accepted
Is generally the hypothesis that is
believed (or needed to be proven) to be
true by the researcher
© 2002 Prentice-Hall, Inc.
Chap 7-6
Hypothesis Testing Process
Assume the
population
mean age is 50.
( H0 : 50)
Identify the Population
Is X 20 likely if ?
No, not likely!
REJECT
Null Hypothesis
© 2002 Prentice-Hall, Inc.
Take a Sample
X 20
Chap 7-7
Reason for Rejecting H0
Sampling Distribution of X
... Therefore,
we reject the
null hypothesis
that m = 50.
It is unlikely that
we would get a
sample mean of
this value ...
... if in fact this were
the population mean.
20
© 2002 Prentice-Hall, Inc.
= 50
If H0 is true
X
Chap 7-8
Level of Significance,
Defines unlikely values of sample statistic if
null hypothesis is true
Is designated by
Called rejection region of the sampling distribution
, (level of significance)
Typical values are .01, .05, .10
Is selected by the researcher at the beginning
Provides the critical value(s) of the test
© 2002 Prentice-Hall, Inc.
Chap 7-9
Level of Significance
and the Rejection Region
H0: 3
H1: < 3
H0: 3
H1: > 3
Rejection
Regions
0
0
H0: 3
H1: 3
Critical
Value(s)
/2
0
© 2002 Prentice-Hall, Inc.
Chap 7-10
Errors in Making Decisions
Type I Error
Rejects a true null hypothesis
Has serious consequences
The probability of Type I Error is
Type II Error
© 2002 Prentice-Hall, Inc.
Called level of significance
Set by researcher
Fails to reject a false null hypothesis
The probability of Type II Error is
The power of the test is 1
Chap 7-11
Errors in Making Decisions
(continued)
Probability of not making Type I Error
1
Called the confidence coefficient
© 2002 Prentice-Hall, Inc.
Chap 7-12
Result Probabilities
H0: Innocent
Jury Trial
Hypothesis Test
The Truth
Verdict
Innocent
Guilty
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The Truth
Innocent Guilty
Correct
Error
Error
Decision H0 True H0 False
Do Not
Reject
H0
Correct Reject
H0
1-
Type II
Error ( )
Type I
Error
( )
Power
(1 - )
Chap 7-13
Type I & II Errors Have an
Inverse Relationship
If you reduce the probability of one
error, the other one increases so that
everything else is unchanged.
© 2002 Prentice-Hall, Inc.
Chap 7-14
Factors Affecting Type II Error
True value of population parameter
Increases when
Increases when increases
Sample size
decreases
Population standard deviation
Increases when the difference between
hypothesized parameter and its true value
decrease
Significance level
Increases when n decreases
n
© 2002 Prentice-Hall, Inc.
Chap 7-15
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
© 2002 Prentice-Hall, Inc.
Chap 7-16
Critical Values
Approach to Testing
Convert sample statistic (e.g.: X ) to test
statistic (e.g.: Z, t or F –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
© 2002 Prentice-Hall, Inc.
Chap 7-17
p-Value Approach to Testing
Convert Sample Statistic (e.g. X ) to Test
Statistic (e.g. Z, t or F –statistic)
Obtain the p-value from a table or computer
p-value: Probability of obtaining a test statistic
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
© 2002 Prentice-Hall, Inc.
If p-value
If p-value
, do not reject H0
, reject H0
Chap 7-18
General Steps in
Hypothesis Testing
e.g.: Test the assumption that the true mean number of of
TV sets in U.S. homes is at least three (
Known)
1. State the H0
H0 : 3
2. State the H1
H1 : 3
3. Choose
=.05
4. Choose n
5. Choose Test
© 2002 Prentice-Hall, Inc.
n 100
Z test
Chap 7-19
General Steps in
Hypothesis Testing
6. Set up critical value(s)
(continued)
Reject H0
7. Collect data
8. Compute test statistic
and p-value
-1.645
100 households surveyed
Z
Computed test stat =-2,
p-value = .0228
9. Make statistical decision Reject null hypothesis
10. Express conclusion
© 2002 Prentice-Hall, Inc.
The true mean number of TV
sets is less than 3
Chap 7-20
One-tail Z Test for Mean
( Known)
Assumptions
Population is normally distributed
If not normal, requires large samples
Null hypothesis has or sign only
Z test statistic
Z
© 2002 Prentice-Hall, Inc.
X X
X
X
/ n
Chap 7-21
Rejection Region
H0: 0
H1: > 0
H0: 0
H1: < 0
Reject H0
Reject H0
0
Z Must Be Significantly
Below 0 to reject H0
© 2002 Prentice-Hall, Inc.
Z
0
Z
Small values of Z don’t
contradict H0
Don’t Reject H0 !
Chap 7-22
Example: One Tail Test
Q. 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.
© 2002 Prentice-Hall, Inc.
368 gm.
H0: 368
H1: > 368
Chap 7-23
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
© 2002 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 7-24
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
© 2002 Prentice-Hall, Inc.
1.50
X
Z
1.50
n
Decision:
Do Not Reject at = .05
Conclusion:
No evidence that true
mean is more than 368
Chap 7-25
p -Value Solution
p-Value is P(Z 1.50) = 0.0668
Use the
alternative
hypothesis
to find the
direction of
the rejection
region.
© 2002 Prentice-Hall, Inc.
P-Value =.0668
1.0000
- .9332
.0668
0
From Z Table:
Lookup 1.50 to
Obtain .9332
1.50
Z
Z Value of Sample
Statistic
Chap 7-26
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
© 2002 Prentice-Hall, Inc.
Chap 7-27
One-tail Z Test for Mean
( Known) in PHStat
PHStat | one-sample tests | Z test for the
mean, sigma known …
Example in excel spreadsheet
© 2002 Prentice-Hall, Inc.
Chap 7-28
Example: Two-Tail Test
Q. 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. Test at the
0.05 level.
© 2002 Prentice-Hall, Inc.
368 gm.
H0: 368
H1: 368
Chap 7-29
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
© 2002 Prentice-Hall, Inc.
.025
0 1.96
1.50
Z
Decision:
Do Not Reject at = .05
Conclusion:
No Evidence that True
Mean is Not 368
Chap 7-30
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
© 2002 Prentice-Hall, Inc.
Chap 7-31
Two-tail Z Test for Mean
( Known) in PHStat
PHStat | one-sample tests | Z test for the
mean, sigma known …
Example in excel spreadsheet
© 2002 Prentice-Hall, Inc.
Chap 7-32
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
If this interval contains the hypothesized mean (368),
we do not reject the null hypothesis.
It does. Do not reject.
© 2002 Prentice-Hall, Inc.
Chap 7-33
t Test: Unknown
Assumption
Population is normally distributed
If not normal, requires a large sample
T test statistic with n-1 degrees of freedom
© 2002 Prentice-Hall, Inc.
X
t
S/ n
Chap 7-34
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
© 2002 Prentice-Hall, Inc.
368 gm.
H0: 368
H1: > 368
Chap 7-35
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
© 2002 Prentice-Hall, Inc.
7
1.80
t35
X 372.5 368
t
1.80
S
15
n
36
Decision:
Do Not Reject at = .01
Conclusion:
No evidence that true
mean is more than 368
Chap 7-36
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
© 2002 Prentice-Hall, Inc.
1.80
Chap 7-37
t Test: Unknown in PHStat
PHStat | one-sample tests | t test for the
mean, sigma known …
Example in excel spreadsheet
© 2002 Prentice-Hall, Inc.
Chap 7-38
Proportion
Involves categorical values
Two possible outcomes
“Success” (possesses a certain characteristic) and
“Failure” (does not possesses a certain
characteristic)
Fraction or proportion of population in the
“success” category is denoted by p
© 2002 Prentice-Hall, Inc.
Chap 7-39
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
© 2002 Prentice-Hall, Inc.
p(1 p)
ps
n
Chap 7-40
Example: Z Test for Proportion
Q. A marketing company
claims that it receives
4% responses from its
mailing. To test this
claim, a random
sample of 500 were
surveyed with 25
responses. Test at the
= .05 significance
level.
© 2002 Prentice-Hall, Inc.
Check:
np 500 .04 20
5
n 1 p 500 1 .04
480 5
Chap 7-41
Z Test for Proportion: Solution
H0: p .04
H1: p .04
Test Statistic:
Z
= .05
n = 500
.025
-1.96
© 2002 Prentice-Hall, Inc.
p 1 p
n
.05 .04
.04 1 .04
500
1.14
Decision:
Critical Values: 1.96
Reject
pS p
Do not reject at = .05
Reject
.025
0 1.96 Z
1.14
Conclusion:
We do not have sufficient
evidence to reject the
company’s claim of 4%
response rate.
Chap 7-42
p -Value Solution
(p Value = 0.2542) ( = 0.05).
Do Not Reject.
p Value = 2 x .1271
Reject
Reject
= 0.05
0
1.14
1.96
Z
Test Statistic 1.14 is in the Do Not Reject Region
© 2002 Prentice-Hall, Inc.
Chap 7-43
Z Test for Proportion in PHStat
PHStat | one-sample tests | z test for the
proportion …
Example in excel spreadsheet
© 2002 Prentice-Hall, Inc.
Chap 7-44
Potential Pitfalls and
Ethical Considerations
Randomize data collection method to reduce
selection biases
Do not manipulate the treatment of human
subjects without informed consent
Do not employ “data snooping” to choose
between one-tail and two-tail test, or to
determine the level of significance
© 2002 Prentice-Hall, Inc.
Chap 7-45
Potential Pitfalls
and Ethical Considerations
(continued)
Do not practice “data cleansing” to hide
observations that do not support a stated
hypothesis
Report all pertinent findings
© 2002 Prentice-Hall, Inc.
Chap 7-46
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
© 2002 Prentice-Hall, Inc.
Chap 7-47
Chapter Summary
(continued)
Performed one-tail and two-tail tests
Performed t test for the mean ( unknown)
Performed Z test for the proportion
Discussed potential pitfalls and ethical
considerations
© 2002 Prentice-Hall, Inc.
Chap 7-48