© 2002 Prentice-Hall, Inc. Chap 7-1
Business Statistics: A First course
4th Edition
Chapter 9Fundamentals of Hypothesis Testing: One-Sample Tests
© 2002 Prentice-Hall, Inc. Chap 7-2
Chapter Topics
Hypothesis testing methodology Z test for the mean ( known) P-value approach to hypothesis testing Connection to confidence interval
estimation One-tail tests T test for the mean ( unknown) Z test for the proportion Potential hypothesis-testing pitfalls and
ethical considerations
© 2002 Prentice-Hall, Inc. Chap 7-3
What is a Hypothesis?
A hypothesis is a claim (assumption)about the populationparameter Examples of parameters
are population meanor proportion
The parameter mustbe identified beforeanalysis
I claim the mean GPA of this class is 3.5!
© 1984-1994 T/Maker Co.
© 2002 Prentice-Hall, Inc. Chap 7-4
The Null Hypothesis, H0
States the assumption (numerical) to be tested e.g.: The average number of TV sets in U.S.
Homes is at least three ( ) Is always about a population parameter
( ), not about a sample statistic ( )
0 : 3H
0 : 3H 0 : 3H X
© 2002 Prentice-Hall, Inc. Chap 7-5
The Null Hypothesis, H0
Begins 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 the “=” sign May or may not be rejected
(continued)
© 2002 Prentice-Hall, Inc. Chap 7-6
The Alternative Hypothesis, H1
Is the opposite of the null hypothesis e.g.: The average number of TV sets in
U.S. homes is less than 3 ( ) Challenges the status quo Never contains the “=” sign May or may not be accepted Is generally the hypothesis that is
believed (or needed to be proven) to be true by the researcher
1 : 3H
© 2002 Prentice-Hall, Inc. Chap 7-7
Hypothesis Testing Process
Identify the Population
Assume thepopulation
mean age is 50.
( )
REJECT
Take a Sample
Null Hypothesis
No, not likely!
X 20 likely if Is ?
0 : 50H
20X
© 2002 Prentice-Hall, Inc. Chap 7-8
Sampling Distribution of
= 50
It is unlikely that we would get a sample mean of this value ...
... Therefore, we reject the
null hypothesis that μ = 50.
Reason for Rejecting H0
20
If H0 is trueX
... if in fact this were the population mean.
X
© 2002 Prentice-Hall, Inc. Chap 7-9
Filling Process
You are in charge of the filling process for the one pound jars of Cheese Whiz
You take a random sample of 36 jars each hour to determine if the filling process is in control
© 2002 Prentice-Hall, Inc.
Steps: State the Null Hypothesis (H0: )
State its opposite, the Alternative Hypothesis (H1: 16) Hypotheses are mutually exclusive &
exhaustive Sometimes it is easier to form the
alternative hypothesis first.
Identify the Problem
© 2002 Prentice-Hall, Inc. Chap 7-11
Decision Rule
Accept H0 if 15.8 oz X oz
Reject H0 if X < 15.8 oz or X > 16.2 oz • Assume a sample of 36 jars tested each hour
• Population standard deviation is .6 oz.
© 2002 Prentice-Hall, Inc. Chap 7-12
Possible Results
States of Nature
Correct Decision
Type II Error(
Correct Decision
Type I Error(
Do Not Reject Ho
Reject H0
ACTIONS
H0 is True = 16
H0 is False
© 2002 Prentice-Hall, Inc. Chap 7-13
Type I Error
X
Reject H0 Reject H0
Do Not Reject H0
16.21615.8
- 2 + 2 Z
.0228.0228.4772 .4772
Z
n
X
36
15.8
16 Z
36
16.2
16
© 2002 Prentice-Hall, Inc. Chap 7-14
Z1
36
15.8
15.9
Z2
36
16.2
15.9
15.8 15.9 16.2 X
Reject H0 Reject H0
Do Not Reject H0
.3413.4987
0 Z
.3413
.4987
.8400 =
© 2002 Prentice-Hall, Inc.
Reduce probability of one error and the other one goes up.
& Have an Inverse Relationship
© 2002 Prentice-Hall, Inc.
True Value of Population Parameter Increases When Difference Between Hypothesized Parameter
& True Value Decreases
Significance Level Increases When Decreases
Population Standard Deviation Increases When Increases
Sample Size n Increases When n Decreases
Factors Affecting Type II Error,
n
© 2002 Prentice-Hall, Inc. Chap 7-17
Hypothesis Testing: Steps
1. State H0 and H1
2. Choose - level of significance
3. Use Information in Steps 1 and 2 to Design a Decision Rule
4. Collect Data
5. Compare Sample Data to Decision Rule and State Conclusion.
© 2002 Prentice-Hall, Inc. Chap 7-18
Setting Up H0 and H1
H0 must contain the equality (=) Either H0 or H1 must precisely represent
what is being tested H0 and H1 must be mutually exhaustive
(must cover all possible outcomes)
© 2002 Prentice-Hall, Inc. Chap 7-19
Level of Significance,
Defines unlikely values of sample statistic if null hypothesis is true Called rejection region of the sampling
distribution Is designated by , (level of
significance) Typical values are .01, .05, .10
Is selected by the researcher at the beginning
Provides the critical value(s) of the test
© 2002 Prentice-Hall, Inc. Chap 7-20
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
© 2002 Prentice-Hall, Inc. Chap 7-21
Example: One Tail Test
Q. Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified to be 15 grams. Test at the 0.05 level.
368 gm.
H0: 368 H1: > 368
X
© 2002 Prentice-Hall, Inc. Chap 7-22
Finding Critical Value: One Tail
Z .04 .06
1.6 .9495 .9505 .9515
1.7 .9591 .9599 .9608
1.8 .9671 .9678 .9686
.9738 .9750
Z0 1.645
.05
1.9 .9744
Standardized Cumulative Normal Distribution Table
(Portion)
What is Z given = 0.05?
= .05
Critical Value = 1.645
.95
1Z
© 2002 Prentice-Hall, Inc. Chap 7-23
Example Solution: One Tail Test
= 0.5
n = 25
Critical Value: 1.645
Decision:
Conclusion:
Do Not Reject at = .05
No evidence that true mean is more than 368
Z0 1.645
.05
Reject
H0: 368 H1: > 368 1.50
XZ
n
1.50
© 2002 Prentice-Hall, Inc. Chap 7-24
p -Value Solution
Z0 1.50
P-Value =.0668
Z Value of Sample Statistic
From Z Table: Lookup 1.50 to Obtain .9332
Use the alternative hypothesis to find the direction of the rejection region.
1.0000 - .9332 .0668
p-Value is P(Z 1.50) = 0.0668
© 2002 Prentice-Hall, Inc. Chap 7-25
p -Value Solution(continued)
01.50
Z
Reject
(p-Value = 0.0668) ( = 0.05) Do Not Reject.
p Value = 0.0668
= 0.05
Test Statistic 1.50 is in the Do Not Reject Region
1.645
© 2002 Prentice-Hall, Inc. Chap 7-26
Example: Two-Tail Test
Q. Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified to be 15 grams. Test at the 0.05 level.
368 gm.
H0: 368
H1: 368
X
© 2002 Prentice-Hall, Inc. Chap 7-27
372.5 3681.50
1525
XZ
n
= 0.05
n = 25
Critical Value: ±1.96
Example Solution: Two-Tail Test
Test Statistic:
Decision:
Conclusion:
Do Not Reject at = .05
No Evidence that True Mean is Not 368Z0 1.96
.025
Reject
-1.96
.025
H0: 368
H1: 368
1.50
© 2002 Prentice-Hall, Inc. Chap 7-28
p-Value Solution
(p Value = 0.1336) ( = 0.05) Do Not Reject.
01.50
Z
Reject
= 0.05
1.96
p Value = 2 x 0.0668
Test Statistic 1.50 is in the Do Not Reject Region
Reject
© 2002 Prentice-Hall, Inc. Chap 7-29
For 372.5, 15 and 25,
the 95% confidence interval is:
372.5 1.96 15 / 25 372.5 1.96 15 / 25
or
366.62 378.38
If this interval contains the hypothesized mean (368),
we do not reject the null hypothesis.
I
X n
t does. Do not reject.
Connection to Confidence Intervals
© 2002 Prentice-Hall, Inc. Chap 7-30
t Test: Unknown
Assumption Population is normally distributed If not normal, requires a large sample
T test statistic with n-1 degrees of freedom
/
XtS n
© 2002 Prentice-Hall, Inc. Chap 7-31
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, ands 15. Test at the 0.01 level.
368 gm.
H0: 368 H1: 368
is not given
© 2002 Prentice-Hall, Inc. Chap 7-32
Example Solution: One-Tail
= 0.01
n = 36, df = 35
Critical Value: 2.4377
Test Statistic:
Decision:
Conclusion:
Do Not Reject at = .01
No evidence that true mean is more than 368t35
0 2.4377
.01
Reject
H0: 368 H1: 368
372.5 3681.80
1536
Xt
Sn
1.80
© 2002 Prentice-Hall, Inc. Chap 7-33
p -Value Solution
01.80
t35
Reject
(p Value is between .025 and .05) ( = 0.01). Do Not Reject.
p Value = [.025, .05]
= 0.01
Test Statistic 1.80 is in the Do Not Reject Region
2.4377
© 2002 Prentice-Hall, Inc. Chap 7-34
PHStat | one-sample tests | t test for the mean, sigma known …
Example in excel spreadsheet
t Test: Unknown in PHStat
Microsoft Excel Worksheet
© 2002 Prentice-Hall, Inc. Chap 7-35
Involves categorical variables Fraction or % of population in a category If two categorical outcomes, binomial distribution
Either possesses or doesn’t possess the characteristic Sample proportion ( p )
Proportions
sizesample
successesofnumber
n
Xp
© 2002 Prentice-Hall, Inc. Chap 7-36
Example: Z Test for Proportion
• Problem: AstraZeneca claims that less than 5% of patients taking Nexium experience an upset stomach. • Approach: To test this claim, a random sample of 500 patients were interviewed. 15 of the patients experienced stomach pain.
• Solution: Test at the = .05 significance level.
© 2002 Prentice-Hall, Inc. Chap 7-37
= .05 n = 500 p = 15/500 = .03
Z Test for Proportion: Solution
H0: .05
H1: .05
Critical Value:
Reject at Reject at = .05Decision:
Conclusion:We do have sufficient evidence to support the claim that less than
5% of patients experience an upset stomach.
Z0
Reject
.05
Test Statistic:
Z p - (1 - )
n
=.03 -.05
.05 (1 - .05)500
= -2.05
© 2002 Prentice-Hall, Inc. Chap 7-38
= .05 n = 500 p = 15/500 = .03
Z Test for Proportion: Solution
H0: .05
H1: .05
Critical Value:
Reject at Reject at = .05Decision:
Conclusion:We do have sufficient evidence to support the claim that less than
5% of patients experience an upset stomach.
Z0
RejectHo
.05
Test Statistic:
Z p - (1 - )
n
=.03 -.05
.05 (1 - .05)500
= -2.05
© 2002 Prentice-Hall, Inc. Chap 7-39
Z Test for Proportion in PHStat
PHStat | one-sample tests | z test for the proportion …
Example in excel spreadsheet
Microsoft Excel Worksheet
© 2002 Prentice-Hall, Inc. Chap 7-40
Controlling and
A bank wishes to open a new branch if average monthly income is at least 4,000. It does not want to open the branch if average monthly income is less than or equal to 3,800.
Assume that
If the bank wants:What sample size should be used?What should the Decision Rule be?
© 2002 Prentice-Hall, Inc. Chap 7-41
H0: 4,00 H1: ≤ 3,800
n = 99Do not Reject H0 if Xc $Reject H0 if Xc < $3,917
2.33 =Xc - 3,800
500 n-1.645 =
Xc - 4,000
500 n
3,800
4,000 XZ
XZ
Xc
-1.645
Xc
2.330
0
Do not reject Ho
Do not reject H0
Reject H0
reject H0
.05 =
.01 =
© 2002 Prentice-Hall, Inc. Chap 7-42
Potential Pitfalls and Ethical Considerations
Randomize data collection method to reduce selection biases
Do not manipulate the treatment of human subjects without informed consent
Do not employ “data snooping” to choose between one-tail and two-tail test, or to determine the level of significance
© 2002 Prentice-Hall, Inc. Chap 7-43
Potential Pitfalls and Ethical Considerations
Do not practice “data cleansing” to hide observations that do not support a stated hypothesis
Report all pertinent findings
(continued)
© 2002 Prentice-Hall, Inc. Chap 7-44
Chapter Summary
Addressed hypothesis testing methodology
Performed Z Test for the mean ( Known)
Discussed p –Value approach to hypothesis
testing
Made connection to confidence interval
estimation
© 2002 Prentice-Hall, Inc. Chap 7-45
Chapter Summary
Performed one-tail and two-tail tests
Performed t test for the mean
( unknown)
Performed Z test for the proportion
Discussed potential pitfalls and ethical
considerations
(continued)