Download Statistics 702/J702: Homework Solutions - Binomial vs. Hypergeometric Experiments and more Assignments Statistics in PDF only on Docsity!
STAT 702/J702 B.Habing Univ. of S.C. (^1)
STAT 702/J
September 2 nd^ , 2004
Instructor: Brian Habing Department of Statistics LeConte 203 Telephone: 803-777- E-mail: [email protected]
STAT 702/J702 B.Habing Univ. of S.C. (^2)
Today
- Homework Solutions
- The Hypergeometric
- Binomial vs. Hypergeometric
STAT 702/J702 B.Habing Univ. of S.C. (^3)
Ch.1 # 38) A child has six blocks, three of which are red and three of which are green. How many patterns can she make by placing them in a line?
What if three white blocks are added?
STAT 702/J702 B.Habing Univ. of S.C. (^4)
Ch. 1 # 42) How many ways can 11 boys on a soccer team be grouped into 4 forwards, 3 midfielders, 3 defenders, and 1 goalie?
STAT 702/J702 B.Habing Univ. of S.C. (^5)
Ch. 1 # 57) Cabinets A, B, and C each have two drawers with one coin per drawer. A has two gold, B has two silver, and C has one gold and one silver.
A cabinet is chosen at random and a drawer is opened showing a silver. What is the chance the other is silver too?
STAT 702/J702 B.Habing Univ. of S.C. (^6)
Last time… Binomial Experiment
- n identical trials
- Each trial has only two possible outcomes (“Success” or “Failure”)
- Probability of “Success” is a constantp for every trial
- Trials are independent
pk^ pn^ k k
n P k n ⎟⎟ − − ⎠
[ successesin trials]= ( 1 )
STAT 702/J702 B.Habing Univ. of S.C. (^10)
Another way of calculating the probability of an event when all sample points are equally probable is:
P(A) = number of sample points in A total number of sample points
STAT 702/J702 B.Habing Univ. of S.C. (^11)
In general, for a population of sizen withk successes and a sample of sizem we get:
m
n
m k
n r k
r
P [ k successesoutof m ]
STAT 702/J702 B.Habing Univ. of S.C. (^12)
Example – Capture/Recapture)
Goal: To estimate the sizen of a population.
Method: “Randomly” capture, tag, and releaser of them. Then “randomly capture”m of them and see how many are tagged.
STAT 702/J702 B.Habing Univ. of S.C. (^13)
Now the probability of a certain number being captured will be hypergeometric!
m
n
m k
n r k
r
P [ k taggedoutof m ]
STAT 702/J702 B.Habing Univ. of S.C. (^14)
The problem is that we knowr,k, andm, but we are looking for n!
Since we can’t findn exactly, we will attempt to estimate it by choosing the value ofn that “seems most likely”. That is, what value ofn would give us the largest probability of observing thek that we did.
STAT 702/J702 B.Habing Univ. of S.C. (^15)
Mathematically then, we need to find then that maximizes
Ifn was continuous we could try taking the derivative with respect to n and setting it equal to zero.
⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛
⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ −
− ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛
= =
m
n
m k
n r k
r
Ln P [ k taggedoutof m ]
