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# Inference - PowerPoint PPT Presentation

Inference. Mary M. Whiteside, Ph.D. Nonparametric Statistics. Two Sides of Inference. Parametric Interval estimation, xbar Hypothesis testing, m 0 Nonparametric Interval estimates, EDF Hypothesis testing, P(X<Y) > P(X>Y). Meaning of Nonparametric. Not about parameters

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Presentation Transcript

### Inference

Mary M. Whiteside, Ph.D.

Nonparametric Statistics

• Parametric

• Interval estimation, xbar

• Hypothesis testing, m0

• Nonparametric

• Interval estimates, EDF

• Hypothesis testing, P(X<Y) > P(X>Y)

• Methods for non-normal distributions

• Methods for ordinal data

• Data Scales

• Nominal, categorical, qualitative

• Ordinal

• Interval

• Ratio - natural zero

• Random sample from a finite population

• Simple

• Stratified

• Cluster

• Inferences are about the finite population

• Audit comprised of a sample from a population of invoices

• Public opinion polls

• QC samples of delivered goods

• Observations of (iid) random variables

• Inferences are about the probability distributions of the random variables

• Weekly average miles per gallon for your new Lexus

• Chi square tests of independence in medical treatment offered men and women

• Effect of female literacy on infant mortality worldwide

• All random variables, by definition, have probability functions (pmf or pdf) and cumulative probability distributions

• Random variables defined on a random sample (Type 1 or 2) are called statistics with probability distributions that are called sampling distributions

• Statistics support both sides of inference

• Estimators - random variables used to create interval estimates

• Test statistics - random variables used to test hypotheses

• Type I sample - subset of invoices where X = sales tax paid on an invoice randomly selected from a finite population

• Xbar is the average sales tax of n randomly selected invoices

• Xbar is an estimator of m, the average sales tax paid for the population of invoices (with standard deviation s)

• Xbar is a test statistic for testing hypotheses

H0: m = m0

• Xbar is a random variable with sampling distribution asymptotically normal as n increases with mean m and standard deviation sn

• Type 2 sample - the complete set of miles per gallon observations made by you since buying your Lexus where X = mpg for your Lexus in a given week

• Xbar is the average mpg for n observations of X

• Xbar is an estimator of the expected value (mX) of the RV X

• Xbar is a test statistic for testing hypotheses

H0: m = m0

• Xbar is a random variable with sampling distribution asymptotically normal as n increases with mean mX and standard deviationsX/n

• If X from a Type 1 sample is regarded as a random variable, then it has the discrete uniform distribution

• Prob [X = x] = 1/N for all x in the population (where the N values of x are assumed to be unique)

• the kth order statistic is the kth smallest observation

• the first order statistic is the smallest observation in a sample

• the nth order statistic is the largest

• Large body of literature on sampling distributions of order statistics

• Definitions

• EDF

• pth sample quantile

• sample mean, variance, and standard deviation

• unbiased estimators (S2 and s2)

• (point estimate - r*standard error of the estimator, point estimate +q*standard error of the point estimate) where r is the a/2 quantile and q is the (1-a/2) quantile from the sampling distribution of the estimator

• r equals -q in symmetric distributions with mean 0 (z = +/- 1.96 or t = +/-2.02581)

• r does not equal -q in skewed distributions such as Chi squared and F

• Parametric procedures - Assumed normal or normal based from the Central Limit Theorem and sample size

• Xbar is approximately normal if n is large

• Xbar is t if X is normal and s is unknown

• Xbar’s distribution is unknown if X’s distribution is unknown and n is small

• Nonparametric distribution-free procedures I.e. the sampling distribution of the statistic (estimator or test statistic) is “free” from the distribution of X

• rank order statistics

• bootstrapped distributions - a/2 and 1-a/2 quantiles

• Exact distributions with approximate models

• Exact distributions with exact models (but usually small samples)

or

• Asymptotic distributions with exact models