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Fundamentals of Hypothesis
Testing: One-Sample Tests
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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
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A hypothesis is an
assumption about the
population parameter.
A parameteris acharacteristic of the
population, like its mean
or variance.
The parameter must beidentified before
analysis.
I assume the mean GPA
of this class is 3.5!
1984-1994 T/Maker Co.
What is a Hypothesis?
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States the Assumption (numerical) to be tested
e.g. The grade point average of juniors is at least 3.0 (H0: 3.0) 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 theStatus Quo
Always contains the = sign
TheNull Hypothesismay or may not berejected.
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Is theopposite of the null hypothesis e.g. The grade pointaverage of juniors is less than 3.0 (H1: < 3.0)
Challenges the Status Quo
Nevercontains the= sign
The Alternative Hypothesis may or may not beaccepted
Is generally the hypothesis that is believed to betruebythe researcher
The Alternative Hypothesis,H1
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Steps: State the Null Hypothesis (H0: 3.0)
State its opposite, the Alternative Hypothesis (H1: 3H0: = 3H1: 3
/2
Critical
Value(s)
RejectionRegions
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Type I Error Reject True Null Hypothesis (False Positive)
Has Serious Consequences
Probability of Type I Error IsCalledLevel of SignificanceSet by researcher
Type II Error
Do Not Reject False Null Hypothesis (False Negative)
Probability of Type II Error Is (Beta)
Errors in Making Decisions
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H0: Innocent
Jury Trial Hypothesis Test
Actual Situation Actual Situation
Verdict Innocent Guilty Decision H0 True H0 False
Innocent Correct ErrorDo Not
Reject
H0
1 - Type IIError ( )
Guilty Error CorrectReject
H0
Type IError( )
Power
(1 - )
Result Possibilities
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Reduce probability ofone error
and theother onegoes up.
& Have an Inverse Relationship
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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
Factors Affecting Type II Error,
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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 IncreasesWhen n Decreases
Factors Affecting Type II Error,
n
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Choice depends on the cost of the error
Choose little type I error when the cost of rejecting the
maintained hypothesis is high A criminal trial: convicting an innocent person
The Exxon Valdise: Causing an oil tanker to sink
Choose large type I error when you have an interest in
changing the status quo A decision in a startup company about a new piece of software
A decision about unequal pay for a covered group.
How to choose between Type I and Type II errors
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Deregulation leads to lower air travel prices. The university discriminates against women faculty
members
Stock prices increase when a firm announces a layoff.
Example Hypotheses: How do you test them?
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Hiring Policy Hypotheses
A recruiter must decide who to hire. This is like forming a
hypothesis about whether or not the individual predicts to be
a good employee. Assume that an employee is either good
or poor.
WHAT ARE THE NULL AND ALTERNATIVE HYPOTHESES?
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Hiring Policy Hypotheses
NULL: THE INDIVIDUAL DOES NOT MEET THE STANDARDS
(MEANZ)
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Hiring Policy Hypotheses
WHAT ARE THE POSSIBLE ERRORS THAT THE RECRUITER
CAN MAKE?
HOW DO THESE RELATE TO TYPE I AND TYPE II ERRORS?
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Hiring Policy Hypotheses
FAILURE TO HIRE A GOOD EMPLOYEE (failure to reject a
false null=type II error)
FAILURE TO REJECT A POOR EMPLOYEE(rejecting a null
when it is really true is type I error)
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Hiring Policy Hypotheses
A positive decision is a decision to reject the null. A false
positive is therefore a type I error (hiring a poor person).
A negative decision is a failure to reject the null. A false
negative is therefore a type II error (not hiring a good
person)
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The addition of more criteria should increase your ability to
distinguish poor candidates => type I error falls
However, more criteria mean that more good employees are cut
accidently => type II error increases.
When do you use more criteria?
Hiring Policy Hypotheses
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Convert Sample Statistic (e.g., ) to test statistic, for
example a Z, t or F-statistic
Compare to Critical value obtained from a table.
If the test statistic falls in the Critical Region, RejectH0;Otherwise Do Not RejectH0
Critical Value of the test statistic
X
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Probability of Obtaining a Test StatisticMore Extreme( or ) than Actual Sample ValueGivenH0 Is True CalledObserved Level of Significance
Smallest Value of a H0 Can Be Rejected
Used toMake Rejection Decision Ifp value , Do Not Reject H0 Ifp value < , Reject H0
p Value Test
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1. StateH0 H0 : 3.02. StateH1 H1 : < 3 . 03. Choose = .054. Choose n n = 100
5. Choose Test: t Test (or p Value)
Hypothesis Testing: Steps
Test the Assumption that the true mean
grade point average of juniors is at least 3.
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6. Set Up Critical Value(s) t = -1.7
7. Collect Data 100 students sampled
8. Compute Test Statistic Computed Test Stat.= -2
(computed P value=.04, two-tailed test)
9. Make Statistical Decision Reject Null Hypothesis
10. Express Decision The true mean grade point is less than 3.0
Hypothesis Testing: Steps
Test the Assumption that grade point average of juniors is at least3.
(continued)
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Z0
RejectH0
Z0
RejectH0
H0: 0 H1: < 0
H0: 0H1: > 0
Must BeSignificantly
Below = 0 Small values dont contradictH0Dont RejectH0!
Rejection Region
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Assumptions Population isnormally distributed
If not normal, onlyslightly skewed& a large sample taken
(Central limit theorem applies)
Parametric test procedure
t test statistic, with n-1 degrees of freedom
t-Test: Unknown
n
SXt
=
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# in sample - number of parameters that must be
estimated before test statistic can be computed.
For a single sample t-test, we must first estimate the
mean before we can estimate the standard deviation.
Once the mean is estimated, n-1 of the values are left
since we know that the nth value is equal to
Degrees of Freedom
f
=
1
1
n
iixxn
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Example: One Tail t-Test
Does an average box of cerealcontain
more than368grams of cereal? A
random sample of36boxes showed
X = 372.5, and = 15. Test at the= 0.01 level.
368 gm.
H0: 368H1: > 368
is not given,
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PH Stat Entries
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80.1
36
15
3685.372=
=
=
nS
Xt
Example Solution: One Tail
P ti
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Involvescategorical variables
Fraction or%of population in a category
Iftwo categorical outcomes, binomial
distribution Either possesses or doesnt possess the characteristic
Sample proportion(ps)
Proportions
sizesample
successesofnumber
n
Xps =
Z test for proportions
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The null hypothesis for the proportion also implies we know the variance,
since the variance is just P times (1-P).
This is a good approximation when the sample size is large.
If the sample size is small, we could use the binomial distribution to compute
the exact p value that a sample of size n would yield a sample proportionps
given the population proportion P. Using the normal approximation is much
easier.
Z test for proportions
E l Z T t f P ti
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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 = .05significancelevel.
Z T t f P ti S l ti
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Z Test for Proportion: Solution
Critical Values: 1.96Decision:
Conclusion:We do not have sufficient
evidence to reject the companys
claim of 4% response rate.Z0
Reject Reject
.025.025
a
= 1.14
Ch t S
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Chapter Summary
Addressed Hypothesis Testing Methodology
Performed t- Test for the Mean ( unknown)Discussed p-Value Approach to Hypothesis TestingPerformed One Tail and Two Tail Tests
Performed Z Test of Hypothesis for the Proportion