1. Reject H o Accept H o Accept H o Reject H o Accept H o Reject H o 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

2. Hypothesis testing on variances: one sample New method reduces variances in product 1.41<1.5; How small is enough? Suppose H o is 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 H o : No significant evidence of reduced variance.

3. Hypothesis testing on variances: two samples Variance unequal in two populations F dist. with 15 and 24 D.F. Use table: Reject H o at α=0.2: Variances are not equal.

4. 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.

5. Wilcoxon Rank-Sum Test m=12 n=15 W= Rank sum W=212

6. For each type of parametric test there’s a non-parametric version. http://www.tufts.edu/~gdallal/npar.htm

7. Statistical data analysis: final notes 1.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.)

8. 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. ZipfZipf’s law: My 381 students http://special.newsroom.msu.edu/back_to_school/index.html

9. Thanks!