In Exercises 1–4 , you are selecting from the set W = { University of ( C ) onnecticut, ( D ) uke, University of ( F ) lorida, University of ( N ) orth Carolina } of teams that have won the national championship in Division 1 men's basketball multiple times since 2000. List all the ways you can make each selection. Select two teams with repetition; order is important. For example, DD is allowed, and CD is not the same as DC.
In Exercises 1–4 , you are selecting from the set W = { University of ( C ) onnecticut, ( D ) uke, University of ( F ) lorida, University of ( N ) orth Carolina } of teams that have won the national championship in Division 1 men's basketball multiple times since 2000. List all the ways you can make each selection. Select two teams with repetition; order is important. For example, DD is allowed, and CD is not the same as DC.
Solution Summary: The author explains that systematic counting is the simplest way in which we count a set to list all possibilities.
In Exercises 1–4, you are selecting from the set
W
=
{
University of
(
C
)
onnecticut,
(
D
)
uke, University of
(
F
)
lorida, University of
(
N
)
orth Carolina
}
of teams that have won the national championship in Division 1 men's basketball multiple times since 2000. List all the ways you can make each selection.
Select two teams with repetition; order is important. For example, DD is allowed, and CD is not the same as DC.
Microsoft Excel snapshot for random sampling: Also note the formula used for the last
column
02
x✓ fx =INDEX(5852:58551, RANK(C2, $C$2:$C$51))
A
B
1
No.
States
2
1
ALABAMA
Rand No.
0.925957526
3
2
ALASKA
0.372999976
4
3
ARIZONA
0.941323044
5
4 ARKANSAS
0.071266381
Random Sample
CALIFORNIA
NORTH CAROLINA
ARKANSAS
WASHINGTON
G7
Microsoft Excel snapshot for systematic sampling:
xfx INDEX(SD52:50551, F7)
A
B
E
F
G
1
No.
States
Rand No. Random Sample
population
50
2
1 ALABAMA
0.5296685 NEW HAMPSHIRE
sample
10
3
2 ALASKA
0.4493186 OKLAHOMA
k
5
4
3 ARIZONA
0.707914 KANSAS
5
4 ARKANSAS 0.4831379 NORTH DAKOTA
6
5 CALIFORNIA 0.7277162 INDIANA
Random Sample
Sample Name
7
6 COLORADO 0.5865002 MISSISSIPPI
8
7:ONNECTICU 0.7640596 ILLINOIS
9
8 DELAWARE 0.5783029 MISSOURI
525
10
15
INDIANA
MARYLAND
COLORADO
The spread of an infectious disease is often modeled using the following autonomous differential equation:
dI
-
- BI(N − I) − MI,
dt
where I is the number of infected people, N is the total size of the population being modeled, ẞ is a constant determining the rate of
transmission, and μ is the rate at which people recover from infection.
Close
a) (5 points) Suppose ẞ = 0.01, N = 1000, and µ = 2. Find all equilibria.
b) (5 points) For the equilbria in part a), determine whether each is stable or unstable.
c) (3 points) Suppose ƒ(I) = d. Draw a phase plot of f against I. (You can use Wolfram Alpha or Desmos to plot the function, or draw the
dt
function by hand.) Identify the equilibria as stable or unstable in the graph.
d) (2 points) Explain the biological meaning of these equilibria being stable or unstable.
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