Reject H o Accept H o

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Left Tailed Right Tailed Two tailed

Reject Ho

Reject Ho

Reject Ho Accept Ho

Accept Ho Reject Ho

Accept Ho

http://library.beau.org/gutenberg/1/0/9/6/10962/10962-h/images/069.png

http://www.pindling.org/Math/Statistics/Textbook/Chapter8_two_population_inference/proportion_independent.htm

Hypothesis testing on variances: one sample

New method reduces variances in product

1.41<1.5; How small is enough?

Suppose Hois true (σ²= 1.5), how likely is it to observe S²≤1.41 ?

Chi-sq. with n-1 D.F.

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

Variance unequal in two populations

F dist. with 15 and 24 D.F.

Use table:

Reject Ho at α=0.2: Variances are not equal.

Non-parametric statistics
• All hypothesis testing so far deals with parametersµ, σof 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 Test

m=12 n=15

Rank sum W=212

W=

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

Statistical data analysis: final notes
• All tests based on T dist. requires normality in original population. When sample size is big (>30), applicable even not normal.
• Tests based on Chi-sq. & F dist. are sensitive to violation of normality. Test of normality.
• Some datasets are normal only after log-transformation.
• Use non-parametric tests when data not normal.
• Watch out for outliers! (box plot helps)
• It never hurts to visualize your data!!
• 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:

My 381 students

http://special.newsroom.msu.edu/back_to_school/index.html