Download Business Statistics: A Decision

Business Statistics:
A Decision-Making Approach
7th Edition
Chapter 9
Introduction to
Hypothesis Testing
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-1
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, 7e © 2008 Prentice-Hall, Inc.
Chap 9-2
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 µ = $42

population proportion
Example: The proportion of adults in this
city with cell phones is π = .68
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-3
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, 7e © 2008 Prentice-Hall, Inc.
H0 : x  3
Chap 9-4
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, 7e © 2008 Prentice-Hall, Inc.
Chap 9-5
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: µ < 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 – a research hypothesis
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-6
Formulating Hypotheses

Example 1: Ford motor company has
worked to reduce road noise inside the cab
of the redesigned F150 pickup truck. It
would like to report in its advertising that
the truck is quieter. The average of the
prior design was 68 decibels at 60 mph.

What is the appropriate hypothesis test?
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-7
Formulating Hypotheses

Example 1: Ford motor company has worked to reduce road noise inside
the cab of the redesigned F150 pickup truck. It would like to report in its
advertising that the truck is quieter. The average of the prior design was
68 decibels at 60 mph.

What is the appropriate test?
H0: µ ≥ 68 (the truck is not quieter) status quo
HA: µ < 68 (the truck is quieter) wants to support

If the null hypothesis is rejected, Ford has sufficient
evidence to support that the truck is now quieter.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-8
Formulating Hypotheses

Example 2: The average annual income of
buyers of Ford F150 pickup trucks is
claimed to be $65,000 per year. An
industry analyst would like to test this
claim.

What is the appropriate hypothesis test?
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-9
Formulating Hypotheses

Example 1: The average annual income of buyers of Ford F150
pickup trucks is claimed to be $65,000 per year. An industry
analyst would like to test this claim.

What is the appropriate test?
H0: µ = 65,000 (income is as claimed) status quo
HA: µ ≠ 65,000 (income is different than claimed)

The analyst will believe the claim unless
sufficient evidence is found to discredit it.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-10
Hypothesis Testing Process
Claim: the population mean age is 50.
Null Hypothesis: H0: µ = 50
Population
Now select a random sample:
Sample
Suppose the sample
mean age is 20:
x = 20
Is x = 20
likely if
µ = 50?
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
If not likely,
REJECT
Null Hypothesis
Reason for Rejecting H0
Sampling Distribution of x
20
If it is unlikely that
we would get a
sample mean of
this value ...
x
μ = 50
If H0 is true
... if in fact this were
the population mean…
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
... then we
reject the null
hypothesis that
μ = 50.
Chap 9-12
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, 7e © 2008 Prentice-Hall, Inc.
Chap 9-13
Errors in Making Decisions
(continued)

Type II Error
 Fail to reject a false null hypothesis
The probability of Type II Error is β

β is a calculated value, the formula is
discussed later in the chapter
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-14
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, 7e © 2008 Prentice-Hall, Inc.
H0 False
Chap 9-15
Type I & II Error Relationship
 Type I and Type II errors cannot 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, 7e © 2008 Prentice-Hall, Inc.
Chap 9-16
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, 7e © 2008 Prentice-Hall, Inc.
The formula used to
compute the value of β
is discussed later in the
chapter
Chap 9-17
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, 7e © 2008 Prentice-Hall, Inc.
Chap 9-18
Hypothesis Tests for the Mean
Hypothesis
Tests for 
σ Known

σ Unknown
Assume first that the population
standard deviation σ is known
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-19
Process of Hypothesis Testing

1. Specify population parameter of interest

2. Formulate the null and alternative hypotheses

3. Specify the desired significance level, α

4. Define the rejection region

5. Take a random sample and determine
whether or not the sample result is in the
rejection region

6. Reach a decision and draw a conclusion
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-20
Level of Significance
and the Rejection Region
Level of significance =
Lower tail test
Example:
H 0: μ ≥ 3
HA: μ < 3

Upper tail test
Two tailed test
Example:
Example:
H 0: μ ≤ 3
HA: μ > 3
H0: μ = 3
HA: μ ≠ 3


-zα 0
Reject H0
Do not
reject H0
0
Do not
reject H0
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
zα
Reject H0
 /2
/2
-zα/2 0
Reject H0
Do not
reject H0
zα/2
Reject H0
Chap 9-21
Critical Value for Lower Tail Test

The cutoff value, -zα or xα ,
H0: μ ≥ 3
HA: μ < 3
is called a critical value

Reject H0
x   μ  z
σ
n
-zα
xα
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Do not reject H0
0
µ=3
Chap 9-22
Critical Value for Upper Tail Test

The cutoff value, zα or xα ,
H0: μ ≤ 3
is called a critical value
HA: μ > 3

Do not reject H0
0
zα
µ=3
xα
x   μ  z
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Reject H0
σ
n
Chap 9-23
Critical Values for
Two Tailed Tests

H0: μ = 3
HA: μ  3
There are two cutoff
values (critical values):
± zα/2
or
xα/2
xα/2
/2
/2
Lower
Reject H0
Upper
Do not reject H0
-zα/2
xα/2
0
µ=3
Lower
x /2  μ  z /2
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Reject H0
zα/2
xα/2
σ
n
Upper
Chap 9-24
The Rejection Region
Lower tail test
Example:
H 0: μ ≥ 3
H A: μ < 3
Upper tail test
Two tailed test
Example:
Example:
H 0: μ ≤ 3
HA: μ > 3
H 0: μ = 3
HA: μ ≠ 3


-zα 0
xα
Reject H0
Do not
reject H0
Reject H0 if z < -zα
i.e., if x < xα
0
zα
xα
Do not
reject H0
Reject H0
Reject H0 if z > zα
i.e., if x > xα
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
 /2
/2
-zα/2 0 zα/2
x α/2(L)
x α/2(U)
Reject H0
Do not
reject H0
Reject H0
Reject H0 if z < -zα/2 or z > zα/2
i.e., if x < xα/2(L) or x > xα/2(U)
Chap 9-25
Two Equivalent Approaches
to Hypothesis Testing

z-units:



For given , find the critical z value(s):

-zα , zα ,or ±zα/2
x μ
Convert the sample mean x to a z test statistic: z 
σ
Reject H0 if z is in the rejection region,
n
otherwise do not reject H0

x units:

Given , calculate the critical value(s)


xα , or xα/2(L) and xα/2(U)
The sample mean is the test statistic. Reject H0 if x is in the
rejection region, otherwise do not reject H0
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-26
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
 H0: μ  3
HA: μ < 3 (This is a lower tail test)
3. Specify the desired level of significance
 Suppose that  = .05 is chosen for this test
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-27
Hypothesis Testing Example
(continued)

4. Determine the rejection region
 = .05
Reject H0
Do not reject H0
-zα= -1.645
0
This is a one-tailed test with  = .05.
Since σ is known, the cutoff value is a z value:
Reject H0 if z < z = -1.645 ; otherwise do not reject H0
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-28
Hypothesis Testing Example

5. Obtain sample evidence and compute the
test statistic
Suppose a sample is taken with the following
results: n = 100, x = 2.84 ( = 0.8 is assumed known)

Then the test statistic is:
x μ
2.84  3  .16
z


 2.0
σ
0.8
.08
n
100
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-29
Hypothesis Testing Example
(continued)

6. Reach a decision and interpret the result
 = .05
z
Reject H0
-1.645
-2.0
Do not reject H0
0
Since z = -2.0 < -1.645, we reject the null
hypothesis that the mean number of TVs in US
homes is at least 3. There is sufficient evidence
that the mean is less than 3.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-30
Hypothesis Testing Example
(continued)

An alternate way of constructing rejection region:
Now
expressed
in x, not z
units
 = .05
x
Reject H0
2.8684
2.84
Do not reject H0
3
σ
0.8
x α  μ  zα
 3  1.645
 2.8684
n
100
Since x = 2.84 < 2.8684,
we reject the null
hypothesis
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-31
p-Value Approach to Testing

Convert Sample Statistic ( x ) to Test Statistic
(a z value, if σ is known)

Determine 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
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-32
p-Value Approach to Testing
(continued)

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
Smallest value of  for which H0 can be
rejected
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-33
p-value example

Example: How likely is it to see a sample mean
of 2.84 (or something further below the mean) if
the true mean is  = 3.0?
P( x  2.84 | μ  3.0)



2.84  3.0 
 P z 

0.8


100


 P(z  2.0)  .0228
 = .05
p-value =.0228
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
2.8684
2.84
-1.645
-2.0
3
x
0
z
Chap 9-34
p-value example
(continued)

Compare the p-value with 

If p-value <  , reject H0

If p-value   , do not reject H0
 = .05
Here: p-value = .0228
 = .05
p-value =.0228
Since .0228 < .05, we reject
the null hypothesis
2.8684
3
2.84
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-35
Example: Upper Tail z Test
for Mean ( Known)
A phone industry manager thinks that
customer monthly cell phone bill have
increased, and now average over $52 per
month. The company wishes to test this
claim. (Assume  = 10 is known)
Form hypothesis test:
H0: μ ≤ 52 the average is not over $52 per month
HA: μ > 52
the average is greater than $52 per month
(i.e., sufficient evidence exists to support the
manager’s claim)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-36
Example: Find Rejection Region
(continued)

Suppose that  = .10 is chosen for this test
Find the rejection region:
Reject H0
 = .10
Do not reject H0
0
zα=1.28
Reject H0
Reject H0 if z > 1.28
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-37
Review:
Finding Critical Value - One Tail
What is z given  = 0.10?
.90
Standard Normal
Distribution Table (Portion)
.10
 = .10
.50 .40
Z
.07
.08
.09
1.1 .3790 .3810 .3830
1.2 .3980 .3997 .4015
z
0 1.28
1.3 .4147 .4162 .4177
Critical Value
= 1.28
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-38
Example: Test Statistic
(continued)
Obtain sample evidence and compute the test
statistic
Suppose a sample is taken with the following
results: n = 64, x = 53.1 (=10 was assumed known)

Then the test statistic is:
x μ
53.1  52
z 

 0.88
σ
10
n
64
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-39
Example: Decision
(continued)
Reach a decision and interpret the result:
Reject H0
 = .10
Do not reject H0
1.28
0
z = .88
Reject H0
Do not reject H0 since z = 0.88 ≤ 1.28
i.e.: there is not sufficient evidence that the
mean bill is over $52
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-40
p -Value Solution
(continued)
Calculate the p-value and compare to 
p-value = .1894
Reject H0
 = .10
0
Do not reject H0
1.28
z = .88
Reject H0
P( x  53.1 | μ  52.0)



53.1  52.0 
 P z 

10


64


 P(z  0.88)  .5  .3106
 .1894
Do not reject H0 since p-value = .1894 >  = .10
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-41
Critical Value
Approach to Testing

When σ is known, convert sample statistic ( x ) to
a z test statistic
Hypothesis
Tests for 
 Known
 Unknown
The test statistic is:
x μ
z 
σ
n
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-42
Critical Value
Approach to Testing

When σ is unknown, convert sample statistic ( x )
to a t test statistic
Hypothesis
Tests for 
 Known
 Unknown
The test statistic is:
t n1
(The population must be
approximately normal)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
x μ

s
n
Chap 9-43
Hypothesis Tests for μ,
σ Unknown

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 (critical values
are from the t-distribution with n-1 d.f.)

5. Obtain sample evidence and compute the
t test statistic

6. Reach a decision and interpret the result
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-44
Example: Two-Tail Test
( Unknown)
The average cost of a
hotel room in New York
is said to be $168 per
night. A random sample
of 25 hotels resulted in
x = $172.50 and
s = $15.40. Test at the
 = 0.05 level.
H0: μ = 168
HA: μ  168
(Assume the population distribution is normal)
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-45
Example Solution: Two-Tail Test
H0: μ = 168
HA: μ  168
  = 0.05
 n = 25
 Critical Values:
t24 = ± 2.0639
  is unknown, so
use a t statistic
/2=.025
Reject H0
-tα/2
-2.0639
t n1 
/2=.025
Do not reject H0
0
Reject H0
tα/2
1.46 2.0639
x μ
172.50  168

 1.46
s
15.40
n
25
Do not reject H0: not sufficient evidence that
true mean cost is different than $168
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-46
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 population in the
“success” category is denoted by π
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-47
Proportions
(continued)

The sample proportion of successes is denoted
by p :


x number of successes in sample
p

n
sample size
When both nπ and n(1- π) are at least 5, p
is approximately normally distributed with mean
and standard deviation

μp  π
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
π(1 π)
σp 
n
Chap 9-48
Hypothesis Tests for Proportions

The sampling
distribution of p is
normal, so the test
statistic is a z
value:
z
pπ
π(1  π)
n
Hypothesis
Tests for π
nπ  5
and
n(1-π)  5
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
nπ < 5
or
n(1-π) < 5
Not discussed
in this chapter
Chap 9-49
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.
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Check:
n π = (500)(.08) = 40

n(1-π) = (500)(.92) = 460
Chap 9-50
Z Test for Proportion: Solution
Test Statistic:
H0: π = .08
HA: π  .08
z
 = .05
n = 500, p = .05
pπ
.05  .08

 2.47
π(1  π)
.08(1  .08)
n
500
Decision:
Critical Values: ± 1.96
Reject
Reject
Reject H0 at  = .05
Conclusion:
.025
.025
-1.96
0
1.96
z
-2.47
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
There is sufficient
evidence to reject the
company’s claim of 8%
response rate.
Chap 9-51
p -Value Solution
(continued)
Calculate the p-value and compare to 
(For a two sided test the p-value is always two sided)
Do not reject H0
Reject H0
/2 = .025
Reject H0
/2 = .025
.0068
.0068
p-value = .0136:
P(z  2.47)  P(x  2.47)
 2(.5  .4932)
 2(.0068)  0.0136
-1.96
z = -2.47
0
1.96
z = 2.47
Reject H0 since p-value = .0136 <  = .05
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-52
Type II Error

Type II error is the probability of
failing to reject a false H0
Suppose we fail to reject H0: μ  52
when in fact the true mean is μ = 50

50
52
Reject
H0: μ  52
Do not reject
H0 : μ  52
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-53
Type II Error
(continued)

Suppose we do not reject H0:   52 when in fact
the true mean is  = 50
This is the range of x where
H0 is not rejected
This is the true
distribution of x if  = 50
50
52
Reject
H0:   52
Do not reject
H0 :   52
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-54
Type II Error
(continued)

Suppose we do not reject H0: μ  52 when
in fact the true mean is μ = 50
Here, β = P( x  cutoff ) if μ = 50
β

50
52
Reject
H0: μ  52
Do not reject
H0 : μ  52
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-55
Calculating β

Suppose n = 64 , σ = 6 , and  = .05
σ
6
 52  1.645
 50.766
n
64
cutoff  x   μ  z 
(for H0 : μ  52)
So β = P( x  50.766 ) if μ = 50

50
50.766
Reject
H0: μ  52
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
52
Do not reject
H0 : μ  52
Chap 9-56
Calculating β
(continued)

Suppose n = 64 , σ = 6 , and  = .05



50.766  50 
P( x  50.766 | μ  50)  P z 
 P(z  1.02)  .5  .3461  .1539

6


64


Probability of
type II error:

β = .1539
50
52
Reject
H0: μ  52
Do not reject
H0 : μ  52
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-57
Hypothesis Tests in PHStat
Options
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-58
Sample PHStat Output
Input
Output
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-59
Chapter Summary

Addressed hypothesis testing methodology

Performed z Test for the mean (σ known)

Discussed p–value approach to
hypothesis testing

Performed one-tail and two-tail tests . . .
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-60
Chapter Summary
(continued)

Performed t test for the mean (σ
unknown)

Performed z test for the proportion

Discussed Type II error and computed its
probability
Business Statistics: A Decision-Making Approach, 7e © 2008 Prentice-Hall, Inc.
Chap 9-61