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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-1
Chapter 9
Fundamentals of HypothesisTesting: One-Sample Tests
Statistics for ManagersUsing Microsoft Excel
5th Edition
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-2
What is a Hypothesis?
A hypothesis is a claim(assumption) about apopulation parameter:
population mean
population proportion
Example: The mean monthly cell phone billof this city is = $42
Example: The proportion of adults in thiscity with cell phones is p = .68
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-3
The Null Hypothesis, H0
States the assumption (numerical) to betested
Example: The average number of TV sets inU.S. Homes is equal to three ( )
Is always about a population parameter,
not about a sample statistic
3:H0 !
3: ! 3X: !
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-4
The Null Hypothesis, H0
Begin with the assumption that the nullhypothesis is true
Similar to the notion of innocent untilproven guilty
Refers to the status quo
Always contains = , or u sign May or may not be rejected
(continued)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-5
The Alternative Hypothesis, H1
Is the opposite of the null hypothesis
e.g., The average number of TV sets in U.S.homes is not equal to 3 ( H1: 3 )
Challenges the status quo
Never contains the = , or u sign
May or may not be accepted
Is generally the hypothesis that theresearcher is trying to show
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc.
Population
Claim: thepopulationmean age is 50.(Null Hypothesis:
REJECT
Supposethe samplemean ageis 20: X = 20
SampleNull Hypothesis
20 likely if = 50?!Is
Hypothesis Testing Process
If not likely,
Now select arandom sample
H0: = 50 )
X
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-7
Sampling Distribution ofX
= 50If H0 is true
If it is unlikely thatwe would get asample mean ofthis value ...
... then wereject the null
hypothesis that = 50.
Reason for Rejecting H0
20
... if in fact this werethe population mean
X
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-8
Level of Significance, E
Defines the unlikely values of the sample
statistic if the null hypothesis is true
Defines rejection region of the samplingdistribution
Is designated by E , (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
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-9
Level of Significanceand the Rejection Region
H0: 3
H1: < 3
0
H0: 3
H1: > 3
E
E
Representscritical value
Lower-tail test
Level of significance = E
0Upper-tail test
Two-tail test
Rejection
region isshaded
/2
0
E/2EH0: = 3
H1: 3
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-10
Errors in Making Decisions
Type IError
Reject a true null hypothesis
Considered a serious type of error
The probability of Type I Error is E
Called level of significance of the test Set by researcher in advance
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-12
Outcomes and Probabilities
Actual
SituationDecision
Do NotReject
H0
No error(1 - )E
Type IIError( )
RejectH0
Type IError( )E
Possible Hypothesis Test Outcomes
H0 FalseH0 True
Key:
Outcome(Probability) No Error( 1 - )
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-13
Type I & II Error Relationship
Type I and Type II errors can not happen atthe 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 ( E ) , then
Type II error probability ( )
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-14
Factors Affecting Type II Error
All else equal,
when the difference between
hypothesized parameter and its true value
when E
when when n
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-15
Critical ValueApproach to Testing
For two tailed test for the mean, known:
Convert sample statistic ( ) to test statistic (Z
statistic )
Determine the critical Z values for a specifiedlevel of significance E from a table orcomputer
Decision Rule: If the test statistic falls in therejection region, reject H0 ; otherwise do not
reject H0
X
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-16
Do not reject H0 Reject H0Reject H0
There are twocutoff values(critical values),
defining theregions ofrejection
Two-Tail Tests
E/2
-Z 0
H0: = 3
H1: { 3
+Z
E/2
Lowercritical
value
Uppercritical
value
3
Z
X
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-17
Review: 10 Steps inHypothesis Testing
1. State the null hypothesis, H0
2. State the alternative hypotheses, H1
3. Choose the level of significance,
4. Choose the sample size, n
5. Determine the appropriate statisticaltechnique and the test statistic to use
6. Find the critical values and determine therejection region(s)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-18
Review: 10 Steps inHypothesis Testing
7. Collect data and compute the test statisticfrom the sample result
8. Compare the test statistic to the criticalvalue to determine whether the test statisticsfalls in the region of rejection
9. Make the statistical decision: Reject H0 if the
test statistic falls in the rejection region 10.Express the decision in the context of the
problem
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-19
Hypothesis Testing Example
Test the claim that the true mean # of TVsets in US homes is equal to 3.
(Assume = 0.8)
1-2. State the appropriate null and alternativehypotheses
H0: = 3 H1: 3 (This is a two tailed test)
3. Specify the desired level of significance
Suppose that E = .05 is chosen for this test
4. Choose a sample size
Suppose a sample of size n = 100 is selected
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-20
2 .0.08
.
00
0.8
2.84
n
!
!
!
!
Hypothesis Testing Example
5. Determine the appropriate technique is known so this is a Z test
6. Set up the critical values
ForE = .05 the critical Z values are 1.96
7. Collect the data and compute the test statistic
Suppose the sample results are
n = 100, X = 2.84 ( = 0.8 is assumed known)
So the test statistic is:
(continued)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-21
Reject H0 Do not reject H0
8. Is the test statistic in the rejection region?
E = .05/2
-Z= -1.96 0Reject H0 ifZ < -1.96 or
Z > 1.96;otherwisedo notreject H0
Hypothesis Testing Example(continued)
E = .05/2
Reject H0
+Z= +1.96
Here, Z = -2.0 < -1.96, so thetest statistic is in the rejectionregion
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-22
9-10. Reach a decision and interpret the result
-2.0
Since Z = -2.0 < -1.96, we reject the null hypothesisand conclude that there is sufficient evidence that themean number of TVs in US homes is not equal to 3
Hypothesis Testing Example(continued)
Reject H0 Do not reject H0
E = .05/2
-Z= -1.96 0
E = .05/2
Reject H0
+Z= +1.96
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-23
p-Value Approach to Testing
p-value: Probability of obtaining a test
statistic more extreme ( oru ) than the
observed sample value given H0 is true
Also called observed level of significance
Smallest value of E for which H0 can berejected
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-24
p-Value Approach to Testing
Convert Sample Statistic (e.g., ) to TestStatistic (e.g., Z statistic )
Obtain the p-value from a table or computer
Compare the p-value with E
If p-value < E , reject H0
If p-value u E , do not reject H0
X
(continued)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-25
.0228
E/2 = .025
p-Value Example
Example: How likely is it to see a sample mean of2.84 (or something further from the mean, in eitherdirection) if the true mean is Q = 3.0?
-1.96 0
-2.0
.02282.0)P(Z
.02282.0)P(Z
!"
!
Z1.96
2.0
X = 2.84 is translatedto a Z score of Z = -2.0
p-value
=.0228 + .0228 = .0456
.0228
E/2 = .025
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-26
Compare the p-value with E
If p-value < E , reject H0
If p-value u E , do not reject H0
Here: p-value = .0456E = .05
Since .0456 < .05, wereject the nullhypothesis
(continued)
p-Value Example
.0228
E/2 = .025
-1.96 0
-2.0
Z1.96
2.0
.0228
E/2 = .025
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc.
Connection to Confidence Intervals
ForX = 2.84, = 0.8 and n = 100, the 95%confidence interval is:
2.6832 2.9968
Since this interval does not contain the hypothesizedmean (3.0), we reject the null hypothesis at E = .05
000.( . ).t
000.( . )-.
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-28
One-Tail Tests
In many cases, the alternative hypothesisfocuses on a particular direction
H0: 3
H1: < 3
H0: 3
H1: > 3
This is a lower-tail test since thealternative hypothesis is focused onthe lower tail below the mean of 3
This is an upper-tail test since thealternative hypothesis is focused onthe upper tail above the mean of 3
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-29
Reject H0 Do not reject H0
There is only one
critical value, since
the rejection area isin only one tail
Lower-Tail Tests
E
-Z 0
H0: 3
H1: < 3
Z
X
Critical value
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-30
Reject H0Do not reject H0
Upper-Tail Tests
E
Z0
H0: 3
H1: > 3 There is only one
critical value, since
the rejection area isin only one tail
Critical value
Z
X
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-31
Example: Upper-Tail Z Testfor Mean (W Known)
A phone industry manager thinks thatcustomer monthly cell phone bill haveincreased, and now average over $52 per
month. The company wishes to test thisclaim. (Assume W = 10 is known)
H0: 52 the average is not over $52 per monthH1: > 52 the average is greater than $52 per month
(i.e., sufficient evidence exists to support themanagers claim)
Form hypothesis test:
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-32
Reject H0Do not reject H0
Suppose that E = .10 is chosen for this test
Find the rejection region:
E = .10
1.280
Reject H0
Reject H0 if Z > 1.28
Example: Find Rejection Region
(continued)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-33
Review:One-Tail Critical Value
Z .07 .09
1.1 .8790 .8810 .8830
1.2.8980 .9015
1.3 .9147 .9162 .9177z 0 1.28
.08
Standard NormalDistribution Table (Portion)What is Z given E = 0.10?
E = .10
Critical Value= 1.28
.90
.8997
.10
.90
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-34
Obtain sample and compute the test statistic
Suppose a sample is taken with the following
results: n = 64, X = 53.1 (W=10 was assumed known)
Then the test statistic is:
n
!
!
!
Example: Test Statistic(continued)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-35
Reject H0Do not reject H0
Example: Decision
E = .10
1.280
Reject H0
Do not reject H0 since Z = 0.88 1.28
i.e.: there is not sufficient evidence that themean bill is over $52
Z = .88
Reach a decision and interpret the result:(continued)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-36
Reject H0
E = .10
Do not reject H0 1.28
0
Reject H0
Z = .88
Calculate the p-value and compare to E(assuming that = 52.0)
(continued)
.
.. )(
/
.3.
3. )X(
!
!u!
u!
u
p-value = .1894
p -Value Solution
Do not reject H0 since p-value = .1894 > E = .10
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-37
Example: Two-Tail Test(W Unknown)
The average cost of ahotel room in New York
is said to be $168 pernight. A random sampleof 25 hotels resulted inX = $172.50 and
S = $15.40. Test at the
E = 0.05 level.(Assume the population distribution is normal)
H0: = 168
H1: { 168
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-38
E= 0.05
n = 25
W is unknown, so
use a t statistic Critical Value:
t24 = 2.0639
Example Solution:Two-Tail Test
Do not reject H0: not sufficient evidence thattrue mean cost is different than $168
Reject H0Reject H0
E/2=.025
-t n-1,/2Do not reject H0
0
E/2=.025
-2.0639 2.0639
1.46
25
15.40
168172.50
n
S
Xt
1n!
!
!
1.46
H0: = 168
H1: { 168
t n-1,/2
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc.
Connection to Confidence Intervals
ForX = 172.5, S = 15.40 and n = 25, the 95%confidence interval is:
172.5 - (2.0639) 15.4/ 25 to 172.5 + (2.0639) 15.4/ 25
166.14 178.86
Since this interval contains the Hypothesized mean (168),we do not reject the null hypothesis at E = .05
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-40
Hypothesis Tests for Proportions
Involves categorical variables
Two possible outcomes
Success (possesses a certain characteristic)
Failure (does not possesses that characteristic)
Fraction or proportion of the population in the
success category is denoted by p
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-41
Proportions
Sample proportion in the success category isdenoted by ps
When both np and n(1-p) are at least 5, ps
can be approximated by a normal distributionwith mean and standard deviation
sizesamplesampleinsuccessesofnumber
nXps !!
p sp !n
p)p(1
sp
!
(continued)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-42
The samplingdistribution of psis approximatelynormal, so the teststatistic is a Zvalue:
Hypothesis Tests for Proportions
n
)p(p
ppZ
s
! 1
np u 5and
n(1-p) u 5
HypothesisTests for p
np < 5or
n(1-p) < 5
Not discussedin this chapter
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-43
An equivalent formto the last slide,but in terms of thenumber ofsuccesses, X:
Z Test for Proportionin Terms of Number of Successes
)p(npnpXZ
!
X u 5and
n-X u 5
HypothesisTests for X
X < 5or
n-X < 5
Not discussedin this chapter
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-44
Example: Z Test for Proportion
A marketing companyclaims that it receives8% responses from itsmailing. To test thisclaim, a random sampleof 500 were surveyedwith 25 responses. Testat the E = .05significance level.
Check:
np = (500)(.08) = 40
n(1-p) = (500)(.92) = 460
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-45
Z Test for Proportion: Solution
E = .05
n = 500, ps = .05
Reject H0 at E = .05
H0: p = .08
H1: p { .08
Critical Values: 1.96
Test Statistic:
Decision:
Conclusion:
z0
Reject Reject
.025.025
1.96
-2.47
There is sufficientevidence to reject thecompanys claim of 8%
response rate.
2.47
500
.08).08(1
.08.05
n
p)p(1
ppZ
s!
!
!
-1.96
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-46
Do not reject H0Reject H0Reject H0
E/2 = .025
1.960
Z = -2.47
Calculate the p-value and compare to E(For a two sided test the p-value is always two sided)
(continued)
0.01362(.0068)
2. )(2. )(
!!
ue
p-value = .0136:
p-Value Solution
Reject H0 since p-value = .0136 < E = .05
Z = 2.47
-1.96
E/2 = .025
.0068.0068
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc.
Deregulation leads to lower air travel prices.
The university discriminates against women
faculty members Stock prices increase when a firm announces a
layoff.
The cost minimizing strategy is to hire high school
graduates via recommendations from our currentemployees.
Example Hypotheses:
How do you test them?
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-48
Hiring Policy Hypotheses
A recruiter must decide who to hire. This is likeforming a hypothesis about whether or not theindividual predicts to be a good employee.
Assume that an employee is either good orpoor.
WHAT ARE THE NULL AND ALTERNATIVE
HYPOTHESES?
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-49
Hiring Policy Hypotheses
NULL: THE INDIVIDUAL DOES NOT MEETTHE STANDARDS (MEANZ)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-50
Hiring Policy Hypotheses
WHAT ARE THE POSSIBLE ERRORS THATTHE RECRUITER CAN MAKE?
HOW DO THESE RELATE TO TYPE I ANDTYPE II ERRORS?
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-51
Hiring Policy Hypotheses
FAILURE TO HIRE A GOOD EMPLOYEE(failure to reject a false null=type II error)
FAILURE TO REJECT A POOREMPLOYEE(rejecting a null when it is reallytrue is type I error)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-52
Hiring Policy Hypotheses
A positive decision is a decision to reject thenull. A false positive is therefore a type I error(hiring a poor person).
A negative decision is a failure to reject thenull. A false negative is therefore a type IIerror (not hiring a good person)
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Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 8-53
Hiring Policy Hypotheses
The addition of more criteria should increaseyour ability to distinguish poor candidates =>type I error falls
However, more criteria mean that more goodemployees are cut accidentally => type IIerror increases.
When do you use more criteria?
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Hiring Policy Hypotheses
When would type II error start to be moreimportant?
What does this tell you about balancing oftype I and type II error and the criteria that areused in hiring different for kinds ofoccupations?