Hypergeometric

1
Hypergeometric

• Sampling with replacement
• Draw a bead from a jar and then replace before
  drawing again
• Sampling without replacement
• Do not replace the bean before the next draw

2
Hypergeometric

• Suppose we draw a sample of 10 mids from the
  Brigade
• Without replacement cant select the same mid
  twice
• However, almost the same as sampling with
  replacement since there are so many mids
• Very unlikely we would have chosen the same mid
  twice anyway

3
Hypergeometric

• Suppose there are 800 women in the Brigade of
  4000
• Draw a sample of 10 mids
• What is the chance of lt3 women?
• Hypergeometric

4
Hypergeometric

• gtgt hgprob(3200,800,10,0,3)
• ans
• 0.8794
• Compare to binomial
• gtgt bprob(10,800/4000,0,3)
• ans
• 0.8791

5
Hypergeometric

• hgprob(G, B, S, lo, hi)
• G Good in the population
• B Bad in the population
• S sample size
• Lo, Hi of those selected that are Good

6
Hypergeometric

• For smaller populations, HG might be more diff
  from binomial
• gtgt hgprob(320,80,10,0,3)
• ans
• 0.8818
• Exercise Find population size so that HG and
  Bino differ by at least 5
• (Keep proportions the same, i.e., 20 Good.)

7
Hypergeometric

• See p. 167 for mean and SD
• If we let r/N-gtp, then mean looks like binomial
• Var looks like binomial except for (N-n)/(N-1)
• Finite population correction factor
• Always lt1

8
Hypergeometric

• Range of HG RV
• Suppose we have a population of size 15, with 9
  Good.
• Draw a sample of 8, then must have at least 2
  good
• Only 6 Bad in the population

9
Hypergeometric

• Suppose we have drawn several items
• Depending on how many Good so far, prob of
  another Good changes
• Jar has 13 balls, 8 Red and 5 White
• Prob(red on first draw) 8/13
• Prob(red on 2nd 1st was red) 7/12
• Prob(red on 2nd 1st was white) 8/12
• Find marginal prob 2nd is red

10
Hypergeometric
11
Hypergeometric
12
Hypergeometric
13
Hypergeometric

• 56/1213 40/1213 96/1213
• 8/13
• Prob(red on 1st draw)
• So marginal probs are not changing, only
  conditionals
• Suppose I draw 2 balls and dont tell you which I
  drew first
• Prob should be the same

14
Hypergeometric

• Inference
• Example 26, p. 169
• Tag and recapture
• Pop size50
• Tags 10 (Good)
• Resample 8
• Prob(3 of the resample have tags)
• gtgt hgprob(10,40,8,3,3)
• ans
• 0.1471

15
Hypergeometric

• Suppose we dont know that there are 50 fish in
  the lake
• Based on 3 of 8 having tags, how many fish might
  there be in the lake without tags?
• For an upper bound on the population, it would be
  unusual to have so many of the 8 with tags
• Find N so that Prob(gt3 of 8 tags) 0.10
• hgprob(10,B,8,3,8)

16
Hypergeometric

• gtgt b40hgprob(10,b,8,3,8)
• ans
• 0.1878
• gtgt b50hgprob(10,b,8,3,8)
• ans
• 0.1203
• gtgt b55hgprob(10,b,8,3,8)
• ans
• 0.0983
• Take 55 as our 90 upper confidence bound

17
Hypergeometric

• For lower bound on pop, it would be unusual to
  have so few tags
• Find N so that Prob(lt3 of 8 tags) 0.10
• gtgt b6hgprob(10,b,8,0,3)
• ans
• 0.0594
• gtgt b7hgprob(10,b,8,0,3)
• ans
• 0.1170

18
Hypergeometric

• Exercise
• Suppose we tag 20 fish and resample 16.
• If 6 have tags, find 90 upper and lower bounds
  for the without tags

19
Hypergeometric

• P-values
• Fishers Exact Test
• Sir Ronald Fisher
• Classify a sample in two ways
• Want to know if the two classifications are
  associated

20
Hypergeometric

• Survey 50 people
• Classify as Male/Female
• Classify as Favor/Against Death Penalty

21
Hypergeometric
22
Hypergeometric
23
Hypergeometric

• H0 Males Females have same opinion on Death
  Penalty
• Ha Males tend to favor more than Females
• PV Prob(17 or more Males favor H0 true)
• If H0 is true, then we have a HG distn

24
Hypergeometric

• G Favor
• B Oppose
• S Males
• Lo, Hi Males who Favor

25
Hypergeometric

• G Favor 22
• B Oppose 28
• S Males 31
• Lo17, Hi 31 (or 22) Males who Favor
• PVhgprob(22, 28, 31, 17, 31)
• 0.0455
• Pretty unlikely if Males and Females are equally
  likely to Favor

26
Hypergeometric

• Exercises
• Repeat this exercise (same Ha), but consider the
  of Females who Oppose
• Suppose 2 of the males had changed from Favor to
  Oppose. Find the PV