Business statistics a decision making approach 6th edition ch08ppln

Business Statistics:
A Decision-Making Approach
6th Edition

Chapter 8
Introduction to
Hypothesis Testing

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-1


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 test statistic, critical value, and
p-value approaches to test the null hypothesis



Know what Type I and Type II errors are



Compute the probability of a Type II error

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-2


What is a Hypothesis?


A hypothesis is a claim
(assumption) about a
population parameter:


population mean
Example: The mean monthly cell phone bill
of this city is  = $42



population proportion
Example: The proportion of adults in this
city with cell phones is p = .68

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-3


The Null Hypothesis, H0


States the assumption (numerical) to be
tested
Example: The average number of TV sets in
U.S. Homes is at least three ( H0 : μ 3 )



Is always about a population parameter,
not about a sample statistic
H0 : μ 3

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

H0 : x 3
Chap 8-4


The Null Hypothesis, H0
(continued)






Begin 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

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-5


The Alternative Hypothesis,
HA


Is the opposite of the null hypothesis


e.g.: The average number of TV sets in U.S.
homes is less than 3 ( HA:  < 3 )



Challenges the status quo



Never contains the “=” , “≤” or “” sign
May or may not be accepted
Is generally the hypothesis that is believed
(or needs to be supported) by the
researcher




Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-6


Hypothesis Testing Process
Claim: the
population
mean age is 50.
(Null Hypothesis:
H0:  = 50 )

Population

Is x20 likely if  = 50?
Suppose
the sample
REJECT

mean age
Null Hypothesis
is 20: x = 20
Business Statistics: A Decision-Making Approach, 6e © 2010 Prentice-

Now select a
random sample

If not likely,

Hall, Inc.

Sample


Reason for Rejecting H0
Sampling Distribution of x

20

x
 = 50
If H0 is true

If it is unlikely that
we would get a
... if in fact this were
sample mean of
the population mean…
this

value
...
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

... then we
reject the null
hypothesis that
 = 50.
Chap 8-8


Level of Significance, 


Defines unlikely values of sample statistic if
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 selected by the researcher at the beginning



Provides the critical value(s) of the test

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-9


Level of Significance
and the Rejection Region
Level of significance =

H0: μ ≥ 3
HA: μ < 3
H0: μ ≤ 3
HA: μ > 3
H0: μ = 3
HA: μ ≠ 3



Represents
critical value



Rejection

region is
shaded

0

Lower tail test


0

Upper tail test

/2
Two tailed test

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

/2
0
Chap 8-10


Errors in Making Decisions


Type I Error
 Reject a true null hypothesis
 Considered a serious type of error
The probability of Type I Error is 
Called level of significance of the test

 Set by researcher in advance


Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-11


Errors in Making Decisions
(continued)


Type II Error
 Fail to reject a false null hypothesis
The probability of Type II Error is β

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-12


Outcomes and Probabilities
Possible Hypothesis Test Outcomes
State of Nature

Key:
Outcome
(Probability)

Decision

H0 True

Do Not
Reject
H0

No error
(1 -  )

Type II Error
(β)

Reject
H0

Type I Error
()

No Error
(1-β)

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

H0 False

Chap 8-13


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 )
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-14


Factors Affecting Type II
Error


All else equal,


β
when the difference between
hypothesized parameter and its true value



β

when





β

when

σ



β

when

n

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-15


Critical Value

Approach to Testing


Convert sample statistic (e.g.: x ) to test
statistic ( Z or t statistic )



Determine the critical value(s) for a specified
level of significance  from a table or computer



If the test statistic falls in the rejection region,
reject H0 ; otherwise do not reject H0

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-16


Lower Tail Tests


H0: μ ≥ 3

The cutoff value,
-zα or xα , is called a
critical value

HA: μ < 3


Reject H0

-zα
xα

x  μ  z 

Do not reject H0

0

μ

σ
n

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-17


Upper Tail Tests


H0: μ ≤ 3

The cutoff value,

HA: μ > 3

zα or xα , is called a
critical value


Do not reject H0

0

zα

μ

xα

x  μ  z 
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Reject H0

σ
n
Chap 8-18


Two Tailed Tests


H0: μ = 3
HA: μ 
3

There are two cutoff
values (critical values):

± zα/2

/2

or

xα/2
xα/2

Lower
Upper

Reject H0

/2
Do not reject H0

-zα/2
xα/2

0

μ0

Lower

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

x /2 μ z /2

Reject H0

zα/2

xα/2

σ
n

Upper

Chap 8-19


Critical Value
Approach to Testing


Convert sample statistic ( x ) to a test statistic
( Z or t statistic )
Hypothesis
Tests for 
 Known

 Unknown

Large
Samples
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Small
Samples
Chap 8-20


Calculating the Test Statistic
Hypothesis
Tests for μ
 Known

 Unknown

The test statistic is:

x μ
z 
σ
n
Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Large
Samples

Small
Samples

Chap 8-21


Calculating the Test Statistic
(continued)

Hypothesis
Tests for 
 Known
The test statistic is:

t n 1

x μ

s
n

But is sometimes
approximated
using a z:

x μ
z 
σ
n

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

 Unknown

Large
Samples

Small
Samples

Chap 8-22


Calculating the Test Statistic
(continued)

Hypothesis
Tests for 
 Known

 Unknown

The test statistic is:

t n 1

x μ

s
n

Large
Samples

Small
Samples

(The population must be
approximately normal)

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-23


Review: Steps in Hypothesis
Testing


1. Specify the population value of interest



2. Formulate the appropriate null and
alternative hypotheses



3. Specify the desired level of significance



4. Determine the rejection region



5. Obtain sample evidence and compute the
test statistic



6. Reach a decision and interpret the result

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-24


Hypothesis Testing Example
Test the claim that the true mean # of TV
sets in US homes is at least 3.
(Assume σ = 0.8)






1. Specify the population value of interest
 The mean number of TVs in US homes

2. Formulate the appropriate null and alternative
hypotheses
 H : μ  3
HA: μ < 3 (This is a lower tail test)
0
3. Specify the desired level of significance
 Suppose that  = .05 is chosen for this test

Business Statistics: A Decision-Making Approach, 6e © 2010 PrenticeHall, Inc.

Chap 8-25