reject h o accept h o accept h o reject h o accept h o reject h o left tailed right tailed two...
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Reject Ho Accept Ho Accept Ho Reject Ho Accept Ho Reject Ho Reject Ho
211
210
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:
H
H
211
210
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:
H
H
211
210
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H
H
Left Tailed Right Tailed Two tailed
http://www.pindling.org/Math/Statistics/Textbook/Chapter8_two_population_inference/proportion_independent.htm
http://library.beau.org/gutenberg/1/0/9/6/10962/10962-h/images/069.png
observedTT P valueP observedTT P valueP
.0 if P2
;0 if P2
P valueP
observedobserved
observedobserved
observed
TTT
TTT
TT
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Hypothesis testing on variances: one sample
.41.1 ,39.3 0,2:data Observed
5.1:
5.1:
2
21
20
SXn
H
H
New method reduces variances in product
1.41<1.5; How small is enough?
Suppose Ho is true (σ²= 1.5), how likely is it to observe S²≤1.41 ?
97.16
)1(
5.1
41.1)120()1(41.1)1()1(41.1
2
2
2
2
2
2
2
2
2
22
Sn
PSn
PnSn
PSP
79.165.1
41.1)120()1( :statisticTest
2
2
2
2
Sn
Chi-sq. with n-1 D.F.
50.79.1625. 219 P
219
50.41.125. 2 SP 50. valueP 25.
Use table:
There’s good chance of observing 1.41 in a random sample, even if the true population variance is 1.5.No reason to reject Ho: No significant evidence of reduced variance.
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Hypothesis testing on variances: two samples
on.distributi an follows ),( trueis Suppose
.100,25;400 ,16:data Observed
:
:
22
21
22
210
222
211
22
211
22
210
FSSH
SnSn
H
H
Variance unequal in two populations
4100
400 :statisticTest
22
21 S
SF dist. with 15 and 24 D.F.
05.108.224,15 FP 05.2 valueP
Use table:
Reject Ho at α=0.2: Variances are not equal.
1,12
21
12
1
222
222
121
211
22
21
21
2
1
1
1
)1(1)1(
)1(1)1(
nnn
n Fn
n
nSn
nSn
S
S
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Non-parametric statistics
• All hypothesis testing so far deals with parameters µ, σ of certain distributions.
• Non-parametric statistics: raw data is converted into ranks. All subsequent analyses are done on these ranks.
• Do not require original data to be normal. • Sum of ranks are approximately normally
distributed.
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Wilcoxon Rank-Sum TestMinutes to heat a room from 60F to 70FHeater A Heater BData (min) Data (min)
69.3 28.656 25.1
22.1 26.447.6 34.953.2 29.848.1 28.423.2 38.513.8 30.252.6 30.634.4 31.860.2 41.643.8 21.1
3637.913.9
Heater A Heater BData (min) rank Data (min) rank
69.3 27 28.6 956 25 25.1 6
22.1 4 26.4 747.6 21 34.9 1553.2 24 29.8 1048.1 22 28.4 823.2 5 38.5 1813.8 1 30.2 1152.6 23 30.6 1234.4 14 31.8 1360.2 26 41.6 1943.8 20 21.1 3
36 1637.9 1713.9 2
Rank sum 212m=12 n=15
W=
Z 12)1(
2)1(
)(
)( :statisticTest
nmmn
nmmW
WVar
WEW
Rank sum W=212
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For each type of parametric test there’s a non-parametric version.
http://www.tufts.edu/~gdallal/npar.htm
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Statistical data analysis: final notes1. All tests based on T dist. requires normality in
original population. When sample size is big (>30), applicable even not normal.
2. Tests based on Chi-sq. & F dist. are sensitive to violation of normality. Test of normality.
3. Some datasets are normal only after log-transformation.
4. Use non-parametric tests when data not normal.5. Watch out for outliers! (box plot helps)6. It never hurts to visualize your data!!7. Yes, you can do it! (Wiki, google, RExcel etc.)
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Power law distribution• Density function:• Word usage, internet, www, city sizes, protein
interactions, income distribution• Active research in physics, computer science,
linguistics, geophysics, sociology, &economics.
Zipf’s law:
kxcxf )(
My 381 students
http://special.newsroom.msu.edu/back_to_school/index.html
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Thanks!