© 1999 Prentice-Hall, Inc. Chap. 8 - 1
Chapter Topics•Hypothesis Testing Methodology
•Z Test for the Mean (Known)
• p-Value Approach to Hypothesis Testing
•Connection to Confidence Interval Estimation
•One Tail Test
• t Test of Hypothesis for the Mean
•Z Test of Hypothesis for the Proportion
© 1999 Prentice-Hall, Inc. Chap. 8 - 2
A hypothesis is an assumption about the population parameter.
A parameter is a Population mean or proportion
The parameter must be identified before analysis.
I assume the mean GPA of this class is 3.5!
© 1984-1994 T/Maker Co.
What is a Hypothesis?
© 1999 Prentice-Hall, Inc. Chap. 8 - 3
• States the Assumption (numerical) to be tested
e.g. The average # TV sets in US homes is at least 3 (H0: 3)
• Begin with the assumption that the null hypothesis is TRUE.
(Similar to the notion of innocent until proven guilty)
The Null Hypothesis, H0
•Refers to the Status Quo•Always contains the ‘ = ‘ sign
•The Null Hypothesis may or may not be rejected.
© 1999 Prentice-Hall, Inc. Chap. 8 - 4
• Is the opposite of the null hypothesise.g. The average # TV sets in US homes
is less than 3 (H1: < 3)
• Challenges the Status Quo
• Never contains the ‘=‘ sign
• The Alternative Hypothesis may or may not be accepted
The Alternative Hypothesis, H1
© 1999 Prentice-Hall, Inc. Chap. 8 - 5
Steps: State the Null Hypothesis (H0: 3) State its opposite, the Alternative
Hypothesis (H1: < 3)Hypotheses are mutually exclusive &
exhaustiveSometimes it is easier to form the
alternative hypothesis first.
Identify the Problem
© 1999 Prentice-Hall, Inc. Chap. 8 - 6
Population
Assume thepopulationmean age is 50.(Null Hypothesis)
REJECT
The SampleMean Is 20
SampleNull Hypothesis
50?20 XIs
Hypothesis Testing Process
No, not likely!
© 1999 Prentice-Hall, Inc. Chap. 8 - 7
Sample Mean = 50
Sampling DistributionIt is unlikely that we would get a sample mean of this value ...
... if in fact this were the population mean.
... Therefore, we reject the null
hypothesis that = 50.
20H0
Reason for Rejecting H0
© 1999 Prentice-Hall, Inc. Chap. 8 - 8
• Defines Unlikely Values of Sample Statistic if Null Hypothesis Is True Called Rejection Region of Sampling
Distribution
• Designated (alpha) Typical values are 0.01, 0.05, 0.10
• Selected by the Researcher at the Start
• Provides the Critical Value(s) of the Test
Level of Significance,
© 1999 Prentice-Hall, Inc. Chap. 8 - 9
Level of Significance, and the Rejection Region
H0: 3
H1: < 30
0
0
H0: 3
H1: > 3
H0: 3
H1: 3
/2
Critical Value(s)
Rejection Regions
© 1999 Prentice-Hall, Inc. Chap. 8 - 10
• Type I Error Reject True Null Hypothesis Has Serious Consequences Probability of Type I Error Is
Called Level of Significance
• Type II Error Do Not Reject False Null Hypothesis Probability of Type II Error Is (Beta)
Errors in Making Decisions
© 1999 Prentice-Hall, Inc. Chap. 8 - 11
H0: Innocent
Jury Trial Hypothesis Test
Actual Situation Actual Situation
Verdict Innocent Guilty Decision H0 True H0 False
Innocent Correct ErrorDo NotReject
H0
1 - Type IIError ( )
Guilty Error Correct RejectH0
Type IError( )
Power(1 - )
Result Possibilities
© 1999 Prentice-Hall, Inc. Chap. 8 - 12
Reduce probability of one error and the other one goes up.
& Have an Inverse Relationship
© 1999 Prentice-Hall, Inc. Chap. 8 - 13
• Convert Sample Statistic (e.g., ) to Standardized Z Variable
• Compare to Critical Z Value(s) If Z test Statistic falls in Critical Region, Reject
H0; Otherwise Do Not Reject H0
Z-Test Statistics (Known)
Test Statistic
X
n
XXZ
X
X
© 1999 Prentice-Hall, Inc. Chap. 8 - 14
• Probability of Obtaining a Test Statistic More Extreme or ) than Actual Sample Value Given H0 Is True
• Called Observed Level of Significance Smallest Value of a H0 Can Be Rejected
• Used to Make Rejection Decision If p value Do Not Reject H0
If p value <, Reject H0
p Value Test
© 1999 Prentice-Hall, Inc. Chap. 8 - 15
1. State H0 H0 : 3
2. State H1 H1 :
3. Choose = .05
4. Choose n n = 100
5. Choose Test: Z Test (or p Value)
Hypothesis Testing: Steps
Test the Assumption that the true mean # of TV sets in US homes is at least 3.
© 1999 Prentice-Hall, Inc. Chap. 8 - 16
6. Set Up Critical Value(s) Z = -1.645
7. Collect Data 100 households surveyed
8. Compute Test Statistic Computed Test Stat.= -2
9. Make Statistical Decision Reject Null Hypothesis
10. Express Decision The true mean # of TV set is less than 3 in the US households.
Hypothesis Testing: Steps
Test the Assumption that the average # of TV sets in US homes is at least 3.
(continued)
© 1999 Prentice-Hall, Inc. Chap. 8 - 17
• Assumptions Population Is Normally Distributed If Not Normal, use large samples Null Hypothesis Has or Sign Only
• Z Test Statistic:
One-Tail Z Test for Mean (Known)
n
xxz
x
x
© 1999 Prentice-Hall, Inc. Chap. 8 - 18
Z0
Reject H0
Z0
Reject H0
H0: H1: < 0
H0: 0 H1: > 0
Must Be Significantly Below = 0
Small values don’t contradict H0
Don’t Reject H0!
Rejection Region
© 1999 Prentice-Hall, Inc. Chap. 8 - 19
Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed X = 372.5. The company has specified to be 15 grams. Test at the 0.05 level.
368 gm.
Example: One Tail Test
H0: 368 H1: > 368
_
© 1999 Prentice-Hall, Inc. Chap. 8 - 20
Z .04 .06
1.6 .5495 .5505 .5515
1.7 .5591 .5599 .5608
1.8 .5671 .5678 .5686
.5738 .5750
Z0
Z = 1
1.645
.50 -.05
.45
.05
1.9 .5744
Standardized Normal Probability Table (Portion)
What Is Z Given = 0.05?
= .05
Finding Critical Values: One Tail
Critical Value = 1.645
© 1999 Prentice-Hall, Inc. Chap. 8 - 21
= 0.025
n = 25
Critical Value: 1.645
Test Statistic:
Decision:
Conclusion:
Do Not Reject at = .05
No Evidence True Mean Is More than 368Z0 1.645
.05
Reject
Example Solution: One Tail
H0: 368 H1: > 368 50.1
n
XZ
© 1999 Prentice-Hall, Inc. Chap. 8 - 22
Z0 1.50
p Value.0668
Z Value of Sample Statistic
From Z Table: Lookup 1.50
.9332
Use the alternative hypothesis to find the direction of the test.
1.0000 - .9332 .0668
p Value is P(Z 1.50) = 0.0668
p Value Solution
© 1999 Prentice-Hall, Inc. Chap. 8 - 23
0 1.50 Z
Reject
(p Value = 0.0668) ( = 0.05). Do Not Reject.
p Value = 0.0668
= 0.05
Test Statistic Is In the Do Not Reject Region
p Value Solution
© 1999 Prentice-Hall, Inc. Chap. 8 - 24
Does an average box of cereal contains 368 grams of cereal? A random sample of 25 boxes showed X = 372.5. The company has specified to be 15 grams. Test at the 0.05 level.
368 gm.
Example: Two Tail Test
H0: 368
H1: 368
© 1999 Prentice-Hall, Inc. Chap. 8 - 25
= 0.05
n = 25
Critical Value: ±1.96
Test Statistic:
Decision:
Conclusion:
Do Not Reject at = .05
No Evidence that True Mean Is Not 368Z0 1.96
.025
Reject
Example Solution: Two Tail
-1.96
.025
H0: 386
H1: 38650.1
2515
3685.372
n
XZ
© 1999 Prentice-Hall, Inc. Chap. 8 - 26
Connection to Confidence Intervals
For X = 372.5oz, = 15 and n = 25,
The 95% Confidence Interval is:
372.5 - (1.96) 15/ 25 to 372.5 + (1.96) 15/ 25
or
366.62 378.38
If this interval contains the Hypothesized mean (368), we do not reject the null hypothesis.
It does. Do not reject.
_
© 1999 Prentice-Hall, Inc. Chap. 8 - 27
Assumptions Population is normally distributed If not normal, only slightly skewed & a large
sample taken
Parametric test procedure
t test statistic
t-Test: Unknown
nSX
t
© 1999 Prentice-Hall, Inc. Chap. 8 - 28
Example: One Tail t-Test
Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, and 15. Test at the 0.01 level.
368 gm.
H0: 368 H1: 368
is not given,
© 1999 Prentice-Hall, Inc. Chap. 8 - 29
Example:Z Test for Proportion
•Problem: A marketing company claims that it receives 4% responses from its Mailing.
•Approach: To test this claim, a random sample of 500 were surveyed with 25 responses.
•Solution: Test at the = .05 significance level.
© 1999 Prentice-Hall, Inc. Chap. 8 - 30
= .05
n = 500
Do not reject at Do not reject at = .05
Z Test for Proportion: Solution
H0: p .04
H1: p .04
Critical Values: 1.96
Test Statistic:
Decision:
Conclusion:We do not have sufficient
evidence to reject the company’s claim of 4% response rate.
Z p - p
p (1 - p)n
s=
.04 -.05.04 (1 - .04)
500
= 1.14
Z0
Reject Reject
.025.025