univariate inferences about a mean
DESCRIPTION
Univariate Inferences about a Mean. Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia. Scenarios. To test if the following statements are plausible - PowerPoint PPT PresentationTRANSCRIPT
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Univariate Inferences Univariate Inferences about a Meanabout a Mean
Shyh-Kang JengShyh-Kang JengDepartment of Electrical Engineering/Department of Electrical Engineering/
Graduate Institute of Communication/Graduate Institute of Communication/
Graduate Institute of Networking and Graduate Institute of Networking and MultimediaMultimedia
ScenariosTo test if the following statements are plausible– A clam by a cram school that their
course can increase the IQ of your children
– A diuretic is effective– An MP3 compressor is with higher
quality– A claim by a lady that she can
distinguish whether the milk is added before making milk tea
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Evaluating Normality of Univariate Evaluating Normality of Univariate Marginal DistributionsMarginal Distributions
sticcharacterith for theon distributi
normal assumedan from departure indicate
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396.1)317.0)(683.0(3683.0ˆeither
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)(in lying data ofportion :ˆ
data, ofsymmetry checkingAfter
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sx,sxp
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Tests of HypothesesDeveloped by Fisher, Pearson, Neyman, etc.Two-sided
One-sided)hypothesis ve(alternati:
)hypothesis (null:
01
00
H
H
)hypothesis ve(alternati:
)hypothesis (null:
01
00
H
H
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Assumption under Null Hypothesis
)1,0(:/
)/,(:
),(:
0
20
20
Nn
XZ
nNX
NX
55
Rejection or Acceptance of Null Hypothesis
66
Student’s t-Statistics
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ii
n
ii
XXn
s
Xn
Xns
Xt
1
22
1
0
1
1
1,
/
77
88
Student’s Student’s tt-distribution-distribution
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2(
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1(
)(
/
f
f
tf
f
f
tf
f
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Student’s Student’s tt-distribution-distribution
99
1010
Student’s Student’s tt-distribution-distribution
Origin of the Name “Student”
Pseudonym of William Gossett at Guinness Brewery in Dublin around the turn of the 20th CenturyGossett use pseudonym because all Guinness Brewery employees were forbidden to publishToo bad Guinness doesn’t run universities
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Test of HypothesisTest of Hypothesis
)2/()()(
,i.e.
)2/(/
if level cesignificanat
offavor in Reject
210
120
2
10
10
n
n
txsxnt
tns
xt
HH
Selection of Often chosen as 0.05, 0.01, or 0.1Actually, Fisher said in 1956:– No scientific worker has a fixed level of
significance at which year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and hid ideas
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Evaluating Normality of Univariate Evaluating Normality of Univariate Marginal DistributionsMarginal Distributions
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),( is ˆ ofon distributi The
),( large, is When
on distributi binomial
:intervalan within samples ofNumber
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pqpN
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yp
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yny
yny
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Confidence Interval for 0
nstXnstXCI
nstXnstX
nstXnstX
nstXnst
tns
X
nn
nn
nn
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n
/)025.0(,/)025.0(:
95.0
/)025.0(/)025.0(Pr
95.0
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95.0
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95.0)025.0(/
Pr
1195
11
11
11
10
1515
Neyman’s Interpretation
01616
Statistical Significance vs. Practical Significance
The cram school claims that its course will increase the IQ of your child statistically significant at the 0.05 levelAssume that 100 students took the courses were tested, and the population standard deviation is 15The actual IQ improvement to be statistically significant at 0.05 level is simply 94.296.15.1)/( 025.0 zn 1717
More Specific HypothesesNull hypothesis
Alternative hypothesis
00 : H
11 : H
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Type I and Type II Errors
1919
Type I and Type II ErrorsTruth
H0
(non-effective)
H1
(effective)
StudyResults
H0
(non-effective)
1- (type II error)
H1
(effective)
(type I error, false alarm)
1-(power)
2020
Power
The probability of concluding that the sample came from the H1 distribution (i.e., concluding there is a significant difference), when it really did from the H1 distribution (there is a difference)
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Power vs. Difference of Means
power
0
1
2222
n/01
Effective SizesHow many samples are required to validate the following claim of the cram school:– Our course will raise IQ levels of your child by 5
points
statistically significant at 0.05 level, and the type II error is 0.1Normal IQ mean is 100, with standard deviation 15Sample standard deviation is assumed to be 15, too
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Effective Sizes
95)(
/
28.1/
105
96.1/
100
as valuecriticalSet
2
zzn
zzn
zn
CV
zn
CV
CV
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