Today: lab 2 due Monday: Quizz 4 Wed: A3 due Friday: Lab 3 due Mon Oct 1: Exam I  this room, 12 pm

Recap last lecture Ch 6.1 Empirical frequency distributions Discrete Continuous Four forms F(Q=k), F(Q=k)/n, F(Qqk), F(Qqk)/n Four uses Summarization gives clue to process Summarization useful for comparisons Used to make statistical decisions Reliability evaluation

Today Read lecture notes!

Distribution of ages of mothers Sample: students that attended class in 1997 Population: MUN students Unknown distribution

Distribution of ages of mothers Sample: students that attended class in 1997 Population: MUN students Unknown distribution Solution: use theoretical frequency dist to characterize pop Assumption: observations are distributed in the same way as theoretical dist Theoretical distribution is a model of a frequency distribution

Commonly used theoretical dist: Discrete Binomial Multinomial Poisson Negative binomial Hypergeometric Uniform Continuous Normal Chi-square (  2) t F Log-normal Gamma Cauchy Weibull Uniform

Commonly used theoretical dist: Discrete Binomial Multinomial Poisson Negative binomial Hypergeometric Uniform Continuous Normal Chi-square (  2) t F Log-normal Gamma Cauchy Weibull Uniform

Theoretical frequency distributions 4 forms Empirical (n=sample) Theoretical (N=pop discrete) Theoretical (N=pop continuous)

Theoretical frequency distributions - 4 uses 1. Clue to underlying process If an empirical dist fits one of the following, this suggests the kind of mechanism that generated the data a)Uniform dist e.g. # of people per table  mechanism: all outcomes have equal prob b)Normal dist e.g. oxygen intake per day  mechanism: several independent factors, no prevailing factor

Theoretical frequency distributions - 4 uses 1. Clue to underlying process c)Poisson dist e.g. # of deaths by horsekick in the Prussian army, per year  mechanism: rare & random event c)Binomial dist e.g. # of heads/tails on coin toss  mechanism: yes/no outcome

Theoretical frequency distributions - 4 uses 2. Summarize data  dist info contained in parameters e.g. number of events per unit space or time can be summarized as the expected value of a Poisson dist

Theoretical frequency distributions - 4 uses 2. Summarize data e.g. number of events per unit space or time can be summarized as the expected value of a Poisson dist Can make comparisons

Theoretical frequency distributions - 4 uses 3. Decision making. Use theoretical dist to calculate p-value

Theoretical frequency distributions - 4 uses 3. Decision making. Use theoretical dist to calculate p-value p(X 1 qx) p(X 2 >x)

Theoretical frequency distributions - 4 uses 3. Decision making. Use theoretical dist to calculate p-value p(X 1 qx) MiniTab: cdf R: pnorm()

Theoretical frequency distributions - 4 uses 4. Reliability. Put probability range around outcome

Theoretical frequency distributions - 4 uses 4. Reliability. Put probability range around outcome MiniTab: invcdf R: qnorm()

Computing probabilities from observed vs theoretical dist Theoretical AdvantagesDisadvantages Easy Assumptions may not apply  wrong p-values FamiliarChecking assumptions is laborious Recipes, known performance Empirical AdvantagesDisadvantages No assumptionsComputation Easy to defendNot always easy to carry out

Ch 6.3 Fit of Observed to Theoretical Will present 2 examples: 1 continuous, 1 discrete More examples in lecture notes

Ch 6.3 Fit of Observed to Theoretical Example 1 (Poisson) Number of coal mining disasters, (England) NDisaster = [ ] sum(N)=47 k = [ ] = outcomes(N) n = 16 observations kF(N=k)

Example 1 (Poisson) Number of coal mining disasters, (England) kF(N=k)F(N=k)/n

Example 1 (Poisson) Number of coal mining disasters, (England) kF(N=k)F(N=k)/nPr(N=k)

Example 1 (Poisson) Number of coal mining disasters, (England) kF(N=k)F(N=k)/nPr(N=k)

Example 1 (Poisson) Number of coal mining disasters, (England) kF(N=k)F(N=k)/nPr(N=k)Obs-Exp

Example 1 (Poisson) Number of coal mining disasters, (England) kF(N=k)F(N=k)/nPr(N=k)Obs-Exp

Example 2 (Normal) Age of mothers of students in quant 1997 Are the ages normally distributed?

Example 2 (Normal) Age of mothers of students in quant 1997 Are the ages normally distributed?

Example 2 (Normal) Age of mothers of students in quant 1997 Are the ages normally distributed? Strategy  work with probability plots  compute cdf

Example 2 (Normal) Age of mothers of students in quant 1997 Are the ages normally distributed? Strategy  work with probability plots  compute cdf Expected distribution:

Example 2 (Normal) Age of mothers of students in quant 1997 Are the ages normally distributed? Strategy  work with probability plots  compute cdf Expected distribution:

Example 2 (Normal) Age of mothers of students in quant 1997 Are the ages normally distributed? Strategy  work with probability plots  compute cdf