Statistics
Friday, May 04, 2012
Today's Agenda:
1. Collect HW #2: Activities 11­6 & 11­9
2. Hand back and go over HW #1: Activities 11­7 & 11­8
3. Activity 11­3: Family Births
4. Activity 11­12: Family Births continued
5. HW #3: none... enjoy PROM if you're going!
Activity 11­3: Family Births
Suppose a couple has two children. Assume that each child is equally likely to be a boy or a girl, regardless of the outcome of previous births.
(a)
Suppose someone argues that the couple is GUARANTEED to have one boy and one girl because that's what a 0.5 probability means: 50% should be boys and 50% should be girls, so 50% of two children is one of each.
Do you believe this argument?
Do you know of any families that have two boys or two girls?
How would you respond to this person to straighten out their reasoning?
Probability is a LONG­TERM property: If you observe a very large number of births, then the proportion who are girls will be very close to one­half. Be careful! In a small number of births, it is possible that the proportion of girls born might not be anywhere near one­half!
(b)
Now, suppose someone else claims that there are three possible outcomes for this family: two boys, two girls, or one of each. Equal likeliness therefore establishes that the probability of each of these outcomes is one­third.
Do you believe this argument?
How would you respond to this person?
We will simulate to approximate the probabilities of gender in families with two children.
(c)
Use the Random Digits Table to simulate the children's genders for four families of two children each.
Assigning Digits:
LABEL: Use single digits. Even digits represent a family having a girl.
Odd digits represent a family having a boy.
TABLE: Starting in Line 101, take two digits at a time to represent one family.
STOP:
Stop when we have four pairs of digits representing four families.
IDENTIFY SAMPLE:
Family #1:
1
9
=
0 girls
Family #2:
2
2
=
2 girls
Family #3:
3
9
=
0 girls
Family #4
5
0
=
1 girl
(d)
Turn to page 627 in your book, select any line you wish and collect data for 20 families. Record the number of girls in each family in the table below:
Family #1:
Family #11
Family #2:
Family #12:
Family #3:
Family #13:
Family #4:
Family #14:
Family #5:
Family #15:
Family #6:
Family #16:
Family #7:
Family #17:
Family #8:
Family #18:
Family #9:
Family #19:
Family #10:
Family #20:
(e)
Based on your (very small) simulation analysis, does it appear that the probability of each of these three outcomes is one­third? Explain.
(f)
How could you obtain better empirical estimates of these probabilities?
(g)
Combine your simulation results with your classmates.
2 girls:
1 girl:
0 girls:
(h)
Based on these (more extensive simulation results, does it appear that the probability of each of these outcomes is one­third? Explain.
Because each of the two children is equally likely to be a boy or girl, the correct way to list the sample space of equally likely outcomes is:
BB
(i)
BG
GB
GG
Use this sample space to determine the theoretical probabilities of gender for a two­child family.
Probability
Are these probabilities reasonably close to the empirical estimates we calculated from the simulations in part (g)?
Activity 11­12: Family Births (continued)
Suppose a couple has FOUR children now. Assume that each child is equally likely to be a boy or a girl, regardless of the outcome of previous births.
(a)
Describe how you would do a simulation to produce approximate probabilities for the various possible gender breakdowns for families with four children.
(b)
Conduct your simulation and record the approximate probabilities (empirical estimates).
(c)
Based on your simulation results, which is more likely: 2 boys + 2 girls or 3 boys + 1 girl / 1 boy + 3 girls ?
(d)
Write out a sample space.
(e)
Calculate the theoretical probabilities for the various possible gender breakdowns:
(f)
Based on your simulation results, which is more likely: 2 boys + 2 girls or 3 boys + 1 girl / 1 boy + 3 girls ?