Transcript Business Statistics: A Decision

Business Statistics:
A Decision-Making Approach
6th Edition
Chapter 8
Introduction to
Hypothesis Testing
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, 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

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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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?
If not likely,
REJECT
Null Hypothesis
Suppose
the sample
mean age
is 20: x = 20
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Now select a
random sample
Sample
Reason for Rejecting H0
Sampling Distribution of x
20
If it is unlikely that
we would get a
sample mean of
this value ...
 = 50
If H0 is true
... if in fact this were
the population mean…
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
x
... 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 © 2005 Prentice-Hall, Inc.
Chap 8-9
Level of Significance
and the Rejection Region
Level of significance =
H0: μ ≥ 3
HA: μ < 3

Represents
critical value

0
Lower tail test
H0: μ ≤ 3
HA: μ > 3

0
Upper tail test
H0: μ = 3
HA: μ ≠ 3
Rejection
region is
shaded
/2
Two tailed test
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, Inc.
Chap 8-16
Lower Tail Tests

H0: μ ≥ 3
The cutoff value,
HA: μ < 3
-zα or xα , is called a
critical value

Reject H0
-zα
xα
x  = μ  z
Do not reject H0
0
μ
σ
n
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, 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
zα
0
μ
x  = μ  z
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Reject H0
xα
σ
n
Chap 8-18
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
μ0
Lower
x /2 = μ  z /2
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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 © 2005 Prentice-Hall, 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
 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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-25
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-26
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-27
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
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 8-28
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-29
p-Value Approach to Testing

Convert Sample Statistic (e.g. x ) to Test
Statistic ( Z or 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
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 8-30
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-31
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, 6e © 2005 Prentice-Hall, Inc.
x
2.8684
2.84
3
Chap 8-32
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-33
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-34
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-35
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-36
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-37
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-38
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-39
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-40
Example Solution: Two-Tail Test
H0: μ = 168
HA: μ  168
  = 0.05
/2=.025
Reject H0
-tα/2
-2.0639
 n = 25
  is unknown, so
use a t statistic
 Critical Value:
t24 = ± 2.0639
t n1 =
/2=.025
Do not reject H0
0
1.46
Reject H0
tα/2
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, 6e © 2005 Prentice-Hall, Inc.
Chap 8-41
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 . . .

Performed t test for the mean (σ unknown)

Discussed type II error and computed its probability
Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.
Chap 8-42