Reject Ho Accept Ho Accept Ho Reject Ho Accept Ho Reject Ho Reject Ho

211

210

:

:

H

H

211

210

:

:

H

H

211

210

:

:

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

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.

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

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.

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

For each type of parametric test there’s a non-parametric version.

http://www.tufts.edu/~gdallal/npar.htm

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.)

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

Thanks!