Statistics Tests

Statistics Tests

1. DIstributions

   • A normal distribution
     • Sample Mean is Test Statistic
       • Unbiased and consistent
       • Normally distributed
     • Sample mean with LT to standard Normal
       • Normally distributed
       • Unbiased and consistent
   • If distribution is unknown or complicated and sample size is large enough we use the CLT
     • Sample mean is ~N(μ,σ2/n)
     • For the binomial distribution
       • X~N(np,np(1-p))
     • For the Poisson Distribution
       • X~N(λ,λ)
   • Sums of normal distributions
     • X^2 Distribution
       • Sum of normal distributions
   • ???
     • F distribution
       • X=(U/m)/(V/n)
   • Samples from a normal distribution with an unknown mean and variance
     • t-distribution
       • T=Z/(sqrt(Y/n))

2. Hypothesis Test for Normal distribution

   • Unknown Variance
     • One sample t-test
       • T= X-μ/S/sqrt n ~tn-1
   • Paired Data
     • Paired t-test
       • D/(s/sqrt(n)) ~ t n-1
       • DIfferences of pairs are independent
   • Two samples
     • Two sample t-test
       • Z=X-Y/σ sqrt (1/n + 1/m) ~ N(0,1)
       • Uses pooled sample
   • Testing equality of variance
     • F-test for equality of variance
       • F=Sx^2/Sy^2 ~ Fm-1,n-1

3. Non-parametric methods

   • Different Underlying Means of two samples
     • Two-sample permutation tests
       • Use Sample Mean as Test Stat
         • Assume: populations differ in location only
       • Mann-Whitney U-test OR Wilcoxon Rank Sun Test
         • Assume independence and at least ordinal data
   • Matched Pairs of data
     • Using the sample mean as the test stat
       • Assume interval and symmetry
     • Wilcoxon Signed Rank Test
       • Independent, Common medium M, symmetric about M
     • Sign Test
       • Assume at least ordinal
   • One-Sample Permutation Test
     • At least ordianal