Sections 3.73.8

1
Sections 3.7-3.8
2
Section 3.7

• Random Sampling
• Definition If n elements from a population are
  drawn in such a way that every set of n elements
  has an equal chance of being selected, the n
  elements are said to be a random sample.

3
Problem 3.76- Page 153

• Suppose you wish to sample n3 elements fro a
  total of N6 elements
  
• a) Count the number of different samples that can
  be drawn.
  
• N a,b,c,d,e,f
• Samples of n3
• 1)a,b,c 2) a,b,d 3) a,b,e 4) a,b,f
• 5) a,c,d 6) a,c,e 7) a,c,f
• 8) a,d,e 9) a,d,f
• 10) a,e,f
• 11) b,c,d 12) b,c,e 13) b,c,f
• 14) b,d,e 15) b,d,f
• 16) b,e,f
• 17) c,d,e 18) c,d,f 19) c,e,f 20) d,e,f

4
3.76 continued

• B) If random sampling is to be employed, what is
  the probability that any particular sample will
  be selected?
  
• C) Show how to use the random number table to
  generate a sample of three elements.
  
• Table I Appendix A
• Row 1 10480 15011 01536 02011
• Row 2 22368 46573 25595 85393

5
Section 3.8

• Some Counting Rules
• Counting Principle (Multiplicative Rule)
• Permutation Formula
• Partitions Rule
• Combination Formula

6

• Counting rules are used to determine the total
  number of outcomes of an experiment(s) or the
  total number of outcomes favorable to an event.
  
• E.g. How many ways can 5 numbers be picked from
  53 numbers and 1 number be picked out of 42
  numbers? (Powerball)
  

7
Multiplicative Rule

• If you can do the 1st experiment in n ways and
  the 2nd experiment in m ways, and the 3rd
  experiment in r ways then you can do all three
  experiments in nmr ways.

8
Permutations Rule

• If you are drawing n elements for a set of N
  elements, then the total number of ways that you
  can do this is
  
• N!
• (N-n)!

9
Partitions Rule
10
Combinations Rule

• N!
• (N-n)!n!

11
When to Use Each Rule.

• With Replacement Multiplicative Rule (Counting
  Principle)
  
• Without replacement
• Order Matters Permutation Rule
• Order Doesnt Matter Partitions Rule
• and Combination Rule

12
Exercises

• Page 163
• 3.85
• An experiment consists os choosing objects
  without regard to order. How many sample points
  are there if you choose the following?
  
• A) 3 objects from 7
• B) 2 objects from 6
• E) r objects from q

13
Exercises

• 3.83
• Use the multiplicative rule to determine the
  number of sample points in the sample space
  corresponding to the experiment of tossing a coin
  the following number of times
• A) 2 times
• B) 3 times
• C) 5 times
• D) n times

14
Exercises

• 3.90
• The 9-digit Zip code has become an integral part
  of the U.S. postal system.
  
• A) How many different Zip codes are available for
  use by the postal service?
  
• B) The first three digits for Chicago Zip code
  are 606. If no other city in the US has these
  first three digits as part of its Zip, how many
  different Zip codes may possibly exist in Chicago?

15
Exercises

• 3.97
• The Rep Governor of a state with no income tax is
  appointing a committee of five members to
  consider changes in the income tax law. There are
  15 representatives available for appointment, 7
  Democrats and 8 republicans. Assume that the Gov
  selects the 5 committee members at random.
• A) How many ways can he select 5 members form 15
  choices?
  
• B) How many ways can he select 5 from 8
  Republicans?
  
• C) What is the probability that no Democrat is
  appointed to the committee? If this were to
  occur, would you conclude that the appointments
  were made at random?
• D) What is the probability that the majority of
  committee members are Republicans? If this were
  to occur, would you conclude that the
  appointments were made at random?