Transcript Business Statistics: A Decision-Making
Chapter 8
Introduction to Hypothesis Testing
Fall 2006 – Fundamentals of Business Statistics 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 Fall 2006 – Fundamentals of Business Statistics 2
Testing Theories
Hypotheses
theories as Competing theories that we want to test about a population are called
Null Hypothesis
(
H Hypotheses
0 ) and in statistics. Specifically, we label these competing
Alternative Hypothesis
(
H
1 or
H A
).
H
0
:
The null hypothesis is the status quo or the prevailing viewpoint.
H A
:
The alternative hypothesis is the competing belief. It is the statement that the researcher is hoping to prove.
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The Null Hypothesis, H
0 (continued) Begin with the assumption that the null hypothesis is true Refers to the status quo Always contains “=” , “≤” or “ ” sign May or may not be rejected Fall 2006 – Fundamentals of Business Statistics 4
The Alternative Hypothesis, H
A Challenges the status quo Never contains the “=” , “≤” or “ ” sign Is generally the hypothesis that is believed (or needs to be supported) by the researcher Provides the “direction of extreme” Fall 2006 – Fundamentals of Business Statistics 5
Hypothesis Testing Process
Claim: the population mean age is 50.
(Null Hypothesis: H 0 :
= 50 ) Population
Now select a random sample
Is
x = 20 likely if = 50?
If not likely, REJECT Null Hypothesis
Fall 2006 – Fundamentals of Business Statistics
Suppose the sample mean age is 20: x = 20 Sample
Deciding Which Theory to Support
Decision making is based on the “rare event” concept. Since the null hypothesis is the status quo, we assume that it is true unless the observed result is extremely unlikely (rare) under the null hypothesis.
Definition
:
If the data were indeed unlikely to be observed under the assumption that H
0
is true, and therefore we reject H
0
in favor of H A , then we say that the data are statistically significant.
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Reason for Rejecting H
0 Sampling Distribution of x Fall 2006 – Fundamentals of Business Statistics
= 50 If H 0 is true x
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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) Is selected by the researcher at the beginning Provides the critical value(s) of the test Fall 2006 – Fundamentals of Business Statistics 9
Level of Significance and the Rejection Region
Level of significance = H 0 : μ ≥ 3 H A : μ < 3 Lower tail test H 0 : μ H A : μ ≤ 3 > 3 Upper tail test H 0 : μ H A : μ = 3 ≠ 3 Two tailed test /2 Fall 2006 – Fundamentals of Business Statistics
0 0 0
/2
Represents critical value
Rejection region is shaded
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Critical Value Approach to Testing
statistic (
Z* or t* s
tatistic ) 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 H 0 ; otherwise do not reject H 0 Fall 2006 – Fundamentals of Business Statistics 11
Critical Value Approach to Testing Convert sample statistic ( ) to a test statistic (
Z* or t* s
tatistic ) Yes Is X ~ N?
No Sample Size?
Small Large (n ≥ 100) Is s known?
Yes No, use sample standard deviation
s
2. Use T~t (n-1) 1. Use Z~N(0,1)
Calculating the Test Statistic: Z
Two-Sided: H
0
≠
μ 0
:
μ
=
μ 0
; H
A
:
μ
One-Sided Upper Tail: H
μ 0
Reject
H 0
if
Z*
>
Z (0.5
−
α
/2) or
Z*
< −
Z (0.5
−
α
/2) , otherwise do not reject
H 0
; H
A
:
μ > μ 0 0
:
μ
≤
One-Sided Lower Tail: H
μ 0
Reject
H 0
if
Z*
do not reject
H 0
>
Z (0.5
−
α
) , otherwise
; H
A
:
μ < μ 0 0
:
μ
≥ Reject
H 0
if
Z*
do not reject
H 0
< -
Z (0.5
−
α
) , otherwise z * = x μ σ n Fall 2006 – Fundamentals of Business Statistics 13
T test Statistic
Two-Sided: H
Reject
H 0
reject
H 0
Reject
H 0
if
0
t
: * *
μ
=
μ
t
1 /
t
0
2
; H One-Sided Upper Tail: H
0
* Reject
H 0
if
t
t
1
One-Sided Lower Tail: H
1
0 A
: :
μ
≠
μ 0
t
μ
1 / ≤ 2
μ 0
; H
, otherwise do not reject
H 0
:
μ
≥
μ 0
; H
A
:
μ < μ 0
, otherwise
A
:
μ > μ 0
do not reject
H 0
t * = x μ s n Fall 2006 – Fundamentals of Business Statistics 14
Review: Steps in Hypothesis Testing
1.
2.
3.
Specify the population value of interest Formulate the appropriate null and alternative hypotheses Specify the desired level of significance 4.
5.
Determine the rejection region Obtain sample evidence and compute the test statistic 6.
Reach a decision and interpret the result Fall 2006 – Fundamentals of Business Statistics 15
Hypothesis Testing Example
Test the claim that the true mean # of TV sets in US homes is less than 3. Assume that s = 0.8
1.
Specify the population value of interest 2.
Formulate the appropriate null and alternative hypotheses 3.
Specify the desired level of significance
Hypothesis Testing Example
4. Determine the rejection region (continued) = Reject H 0 Do not reject H 0 0 Reject H 0 if Z* test statistic < otherwise do not reject H 0 Fall 2006 – Fundamentals of Business Statistics 17
Hypothesis Testing Example
5. Obtain sample evidence and compute the test statistic A sample is taken with the following results: n = 100, x = 2.84
( s = 0.8 is assumed known) Then the test statistic is: Z * = x σ μ = n Fall 2006 – Fundamentals of Business Statistics 18
Hypothesis Testing Example
(continued) 6. Reach a decision and interpret the result = Reject H 0 Do not reject H 0
0
Since Z* = -2.0 < , Fall 2006 – Fundamentals of Business Statistics
z
19
p-Value Approach to Testing
p-value: Probability of obtaining a test statistic more extreme than the observed sample value given H 0 is true Also called observed level of significance Smallest value of
rejected
for which H 0
can be
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p-Value Approach to Testing
Convert Sample Statistic 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 H 0 If p-value , do not reject H 0 Fall 2006 – Fundamentals of Business Statistics 21
P-Value Calculation
Z test statistic Two-Sided: 2 ×min {P(Z ≥ Z*,Z ≤ Z*)} One Sided Upper Tail P(Z ≥ Z*) One-Sided Lower Tail
P
(
Z
≤
Z
*) T test statistic Two-Sided: 2 ×min {P(t ≥ t*,t ≤ t*)} One Sided Upper Tail P(t ≥ t*) One-Sided Lower Tail
P
(
t
≤
t
*) Fall 2006 – Fundamentals of Business Statistics 22
p-value example
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Example: Upper Tail z Test for Mean (
s
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 s = 10 is known) Form hypothesis test: H 0 : μ ≤ 52 the average is not over $52 per month H A : μ > 52 the average
is
greater than $52 per month (i.e., sufficient evidence exists to support the manager’s claim) Fall 2006 – Fundamentals of Business Statistics 24
Example: Find Rejection Region
(continued) Do not reject H 0 0
Reject H 0
=
Reject H 0 Fall 2006 – Fundamentals of Business Statistics 25
Example: Test Statistic
(continued) Obtain sample evidence and compute the test statistic A sample is taken with the following results: n = 64, x = 53.1 ( s =10 was assumed known) Then the test statistic is: Z * = x σ μ n = Fall 2006 – Fundamentals of Business Statistics 26
Example: Decision
(continued) Reach a decision and interpret the result:
Reject H 0
=
Do not reject H 0 0 Reject H 0 Fall 2006 – Fundamentals of Business Statistics 27
p -Value Solution
Calculate the p-value and compare to (continued) 0 Do not reject H 0 Reject H 0 Fall 2006 – Fundamentals of Business Statistics 28
Example: Two-Tail Test (
s
Unknown)
The average cost of a hotel room in New York is said to be $168 per night. A random sample of 25 hotels s = $15.40. Test at the = 0.05 level.
(Assume the population distribution is normal)
H 0 : H A : μ = 168 μ
168
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Outcomes and Probabilities
Key: Outcome (Probability) Possible Hypothesis Test Outcomes Decision
Do Not Reject
H 0
Reject
H 0 State of Nature
H 0 True H 0 False
No error (1 -
) Type I Error
Type II Error ( β ) No Error ( 1 β )
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