Post on 26-Mar-2015
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(c) 2001, Ron S. Kenett, Ph.D. 1
Parametric Statistical Inference
Instructor: Ron S. KenettEmail: ron@kpa.co.il
Course Website: www.kpa.co.il/biostatCourse textbook: MODERN INDUSTRIAL STATISTICS,
Kenett and Zacks, Duxbury Press, 1998
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Course Syllabus
•Understanding Variability•Variability in Several Dimensions•Basic Models of Probability•Sampling for Estimation of Population Quantities•Parametric Statistical Inference•Computer Intensive Techniques•Multiple Linear Regression•Statistical Process Control•Design of Experiments
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Null Hypotheses H0: Put here what is typical of the
population, a term that characterizes “business as usual” where nothing out of the ordinary occurs.
Alternative Hypotheses H1: Put here what is the challenge, the
view of some characteristic of the population that, if it were true, would trigger some new action, some change in procedures that had previously defined “business as usual.”
Definitions
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Step 1.A claim is made.
A new claim is asserted that challenges existing thoughts about a population characteristic.
Suggestion: Form the alternative hypothesis first, since it embodies the challenge.
The Logic of Hypothesis Testing
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The Logic of Hypothesis Testing
Step 2.How much error are you willing to accept?
Select the maximum acceptable error,. The decision maker must elect how much error he/she is willing to accept in making an inference about the population. The significance level of the test is the maximum probability that the null hypothesis will be rejected incorrectly, a Type I error.
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The Logic of Hypothesis Testing
Step 3.If the null hypothesis were true, what would you expect to see?
Assume the null hypothesis is true. This is a very powerful statement. The test is always referenced to the null hypothesis.Form the rejection region, the areas in which the decision maker is willing to reject the presumption of the null hypothesis.
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The Logic of Hypothesis Testing
Step 4.What did you actually see?
Compute the sample statistic. The sample provides a set of data that serves as a window to the population. The decision maker computes the sample statistic and calculates how far the sample statistic differs from the presumed distribution that is established by the null hypothesis.
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The Logic of Hypothesis Testing
Step 5.Make the decision.
The decision is a conclusion supported by evidence. The decision maker will: reject the null hypothesis if the
sample evidence is so strong, the sample statistic so unlikely, that the decision maker is convinced H1 must be true.
fail to reject the null hypothesis if the sample statistic falls in the nonrejection region. In this case, the decision maker is not concluding the null hypothesis is true, only that there is insufficient evidence to dispute it based on this sample.
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The Logic of Hypothesis Testing
Step 6.What are the implications of the decision for future actions?
State what the decision means in terms of the research program.The decision maker must draw out the implications of the decision. Is there some action triggered, some change implied? What recommendations might be extended for future attempts to test similar hypotheses?
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Type I Error: Saying you reject H0 when it really is
true. Rejecting a true H0.
Type II Error: Saying you do not reject H0 when it
really is false. Failing to reject a false H0.
Two Types of Errors
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What are acceptable error levels?
Decision makers frequently use a 5% significance level. Use = 0.05. An -error means that we will decide to
adjust the machine when it does not need adjustment.
This means, in the case of the robot welder, if the machine is running properly, there is only a 0.05 probability of our making the mistake of concluding that the robot requires adjustment when it really does not.
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Three Types of Tests
Nondirectional, two-tail test: H1: pop parameter n.e. value
Directional, right-tail test: H1: pop parameter value
Directional, left-tail test: H1: pop parameter value
Always put hypotheses in terms of population parameters and have H0: pop parameter = value
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Two tailed test
–z +z
Do NotReject H 0
00 Reject HReject H
H0: pop parameter = valueH1: pop parameter n.e. value
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Right tailed test
H0: pop parameter valueH1: pop parameter > value
+z
Do Not Reject H 00 Reject H
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Left tailed test
H0: pop parameter valueH1: pop parameter < value
–z
Do Not Reject H 0Reject H0
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H1
Ho
Ho H1
OKOK
OKOK
TypeType IIErrorError
TypeType IIIIErrorError
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What Test to Apply?
Ask the following questions: Are the data the result of a
measurement (a continuous variable) or a count (a discrete variable)?
Is known? What shape is the distribution of the
population parameter? What is the sample size?
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Test of µ, Known, Population Normally Distributed
Test Statistic:
where is the sample statistic. µ0 is the value identified in the null
hypothesis. is known. n is the sample size.
n
xz 0
–
x
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Test of µ, Known, Population Not Normally Distributed
If n 30, Test Statistic:
If n < 30, use a distribution-free test.
n
xz 0
–
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Test of µ, Unknown, Population Normally Distributed
Test Statistic:
where is the sample statistic. µ0 is the value identified in the null
hypothesis. is unknown. n is the sample size degrees of freedom on t are n – 1.
x
x–
nst 0
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Test of µ, Unknown, Population Not Normally Distributed
If n 30, Test Statistic:
If n < 30, use a distribution-free test.
tx –
0sn
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If both n 5 and n(1 – ) 5,Test Statistic:
where p = sample proportion 0 is the value identified in the null
hypothesis. n is the sample size.
zp–
0
0(1–
0)
n
Test of , Sample Sufficiently Large
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Test of , Sample Not Sufficiently Large
If either n < 5 or n(1 – ) < 5, convert the proportion to the underlying binomial distribution.
Note there is no t-test on a population proportion.
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Observed Significance Levels
A p-Value is: the exact level of significance of the test
statistic. the smallest value can be and still allow us to
reject the null hypothesis. the amount of area left in the tail beyond the test
statistic for a one-tailed hypothesis test or twice the amount of area left in the tail beyond
the test statistic for a two-tailed test. the probability of getting a test statistic from
another sample that is at least as far from the hypothesized mean as this sample statistic is.
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Observed Significance Levels
A p-Value is: the exact level of significance of the test
statistic. the smallest value can be and still allow us to
reject the null hypothesis. the amount of area left in the tail beyond the test
statistic for a one-tailed hypothesis test or twice the amount of area left in the tail beyond
the test statistic for a two-tailed test. the probability of getting a test statistic from
another sample that is at least as far from the hypothesized mean as this sample statistic is.
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Several Samples
Independent Samples: Testing a
company’s claim that its peanut butter contains less fat than that produced by a competitor.
Dependent Samples: Testing the
relative fuel efficiency of 10 trucks that run the same route twice, once with the current air filter installed and once with the new filter.
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Test of (µ1 – µ2), 1 = 2, Populations Normal
Test Statistic
where degrees of freedom on t = n1 + n2 – 2
2–21
22
)1–2
( 21
)1–1
( 2 where
21
112
]2
–1
[– ]2
–1
[
nn
snsnps
nnps
xxt
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The mean of population 1 is equal to the mean of population 2The mean of population 1 is equal to the mean of population 2
(1) Both distributions are normal1 = 2(1) Both distributions are normal1 = 2
HypothesisHypothesis
AssumptionAssumption
Test StatisticTest Statistic
t distribution with df = n1+ n2-2t distribution with df = n1+ n2-2
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222
1121
21
nnsnsnnn
XXt
H0: pop1 = pop2
H1: pop1 n.e. pop2
Example:Comparing Two populations
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-5 0 5
0.0
0.1
0.2
0.3
0.4
0.5
t
t(x;
nu)
nu=5
nu=50
-5 0 5
0.0
0.1
0.2
0.3
0.4
0.5
t
t(x;
nu)
nu=5
nu=50
t distribution with df = n1+ n2-2t distribution with df = n1+ n2-2
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21
nnsnsnnn
XXt
RejectionRejectionRegionRegion
RejectionRejectionRegionRegion
RejectionRejectionRegionRegion
RejectionRejectionRegionRegion
Example:Comparing Two populations
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Test of (µ1 – µ2), 1 n.e. 2, Populations Normal, large n
Test Statistic
with s12 and s2
2 as estimates for 12 and
22
z [x
1– x
2]–[
1–
2]0
s12
n1
s2
2n2
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Test of Dependent Samples(µ1 – µ2) = µd
Test Statistic
where d = (x1 – x2)
= d/n, the average difference
n = the number of pairs of observations
sd = the standard deviation of d
df = n – 1
nd
sdt
d
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Test of (1 – 2), where n1p15, n1(1–p1)5, n2p25, and n2 (1–p2 )
Test Statistic
where p1 = observed proportion, sample 1
p2 = observed proportion, sample 2
n1 = sample size, sample 1
n2 = sample size , sample 2p
n1
p1
n2
p2
n1
n2
zp p
p p n n
1 2
1 11
12
( )
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Test of Equal Variances
Pooled-variances t-test assumes the two population variances are equal.
The F-test can be used to test that assumption.
The F-distribution is the sampling distribution of s1
2/s22 that would
result if two samples were repeatedly drawn from a single normally distributed population.
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Test of 12 = 2
2
If 12 = 2
2 , then 12/2
2 = 1. So the hypotheses can be worded either way.
Test Statistic: whichever is
larger The critical value of the F will be F(/2, 1, 2) where = the specified level of
significance1 = (n – 1), where n is the size of
the sample with the larger variance2 = (n – 1), where n is the size of the sample
with the smaller variance
21
22 or
22
21
s
s
s
sF
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Confidence Interval for (µ1 – µ2)
The (1 – )% confidence interval for the difference in two means: Equal variances, populations normal
Unequal variances, large samples
׳
2
1
1
122
)2
–1
(nnpstxx
2
22
1
21
2 )
2–
1(
n
s
n
szxx ׳
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Confidence Interval for (1 – 2)
The (1 – )% confidence interval for the difference in two proportions:
when sample sizes are sufficiently large.
(p1
– p2
) z2׳
p1(1– p
1)
n1
p2
(1– p2
)
n2
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The mean of population 1 is equal to the mean of population 2The mean of population 1 is equal to the mean of population 2
(1) Both distributions are normal1 = 2(1) Both distributions are normal1 = 2
HypothesisHypothesis
AssumptionAssumption
Test StatisticTest Statistic
The standard deviation of population 1 is equal to the standard deviation of population 2The standard deviation of population 1 is equal to the standard deviation of population 2
Both distributions are normalBoth distributions are normal
The proportion of error in population 1 is equal to the proportion of errors in population 2The proportion of error in population 1 is equal to the proportion of errors in population 2
n1p1 and n2p2 > 5 (approximation by normal distribution)
n1p1 and n2p2 > 5 (approximation by normal distribution)
F distribution with df2 = n1-1 and df2 = n2-1
F distribution with df2 = n1-1 and df2 = n2-1
22
21
s
sF
t distribution with df = n1+ n2-2t distribution with df = n1+ n2-2
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222
1121
21
nnsnsnnn
XXt
Z - Normal distributionZ - Normal distribution
21
2211
/1/11
//
nnpp
nXnXZ
avgavg
21
21
nn
XXpavg
Summary