chap08.pptx

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

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

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

 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 

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)

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

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

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

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

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

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

 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

Example
The Truth
Verdict
Innocent No
error
Type II Error
Guilty Type I Error
Possible Jury Trial Outcomes
Guilty
Innocent
No Error

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 )

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 )

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

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)

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

Hypothesis Tests for the Mean
 Known  Unknown
Hypothesis
Tests for 

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

 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 







(continued)
X

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
(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

 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
(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

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

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


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


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

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

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)

 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

 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

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


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

Using PHStat
Options

Sample PHStat Output
Input
Output

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

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

Answer Sheet for All Problems
 ___________ Null Hypothesis
 ___________ Alternate Hypothesis
 ___________ Alpha
 ___________ p-value
 ___________ Decision (reject or do not reject)
 Conclusion:

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