Practice
A research was interested in the relation between stress and humor. Below
are data from 8 subjects who completed tests of these two traits. Are these
two variables related to each other? How much stress would a person
probably experience if they had no sense of humor (i.e., score = 0)? How
about if they had a high level of humor (i.e., score = 15)?
Stress
4
10
12
5
7
6
2
14
Sense of Humor
2
8
11
3
8
7
3
13
Practice
• r = .91
• Y = .77 + .98(Humor)
• .77 = .77 + .98(0)
• 15.47 = .77 + .98(15)
• You don’t want to have a sense of humor
5
Frequency
4
3
2
1
0
Ace
2
3
4
5
6
7
Card
8
9
10
J
Q
K
What is the probability of picking an ace?
5
Frequency
4
3
2
1
0
Ace
2
3
4
5
6
7
Card
8
9
10
J
Q
K
Probability =
5
Frequency
4
3
2
1
0
Ace
2
3
4
5
6
7
Card
8
9
10
J
Q
K
What is the probability of picking an ace?
4 / 52 = .077 or 7.7 chances in 100
5
Frequency
4
3
2
1
0
Ace
2
3
4
5
6
7
Card
8
9
10
J
Q
K
Card
K (.077)
Q (.077)
J (.077)
10 (.077)
9 (.077)
8 (.077)
7 (.077)
6 (.077)
5 (.077)
4 (.077)
3 (.077)
2 (.077)
Ace (.077)
Frequency
Every card has the same probability of being
picked
5
4
3
2
1
0
Card
K (.077)
Q (.077)
J (.077)
10 (.077)
9 (.077)
8 (.077)
7 (.077)
6 (.077)
5 (.077)
4 (.077)
3 (.077)
2 (.077)
Ace (.077)
Frequency
What is the probability of getting a 10, J, Q, or
K?
5
4
3
2
1
0
Card
K (.077)
Q (.077)
J (.077)
10 (.077)
9 (.077)
8 (.077)
7 (.077)
6 (.077)
5 (.077)
4 (.077)
3 (.077)
2 (.077)
Ace (.077)
Frequency
(.077) + (.077) + (.077) + (.077) = .308
16 / 52 = .308
5
4
3
2
1
0
Card
K (.077)
Q (.077)
J (.077)
10 (.077)
9 (.077)
8 (.077)
7 (.077)
6 (.077)
5 (.077)
4 (.077)
3 (.077)
2 (.077)
Ace (.077)
Frequency
What is the probability of getting a 2 and then
after replacing the card getting a 3 ?
5
4
3
2
1
0
Card
K (.077)
Q (.077)
J (.077)
10 (.077)
9 (.077)
8 (.077)
7 (.077)
6 (.077)
5 (.077)
4 (.077)
3 (.077)
2 (.077)
Ace (.077)
Frequency
(.077) * (.077) = .0059
5
4
3
2
1
0
Card
K (.077)
Q (.077)
J (.077)
10 (.077)
9 (.077)
8 (.077)
7 (.077)
6 (.077)
5 (.077)
4 (.077)
3 (.077)
2 (.077)
Ace (.077)
Frequency
What is the probability that the two cards you
draw will be a black jack?
5
4
3
2
1
0
10 Card = (.077) + (.077) + (.077) + (.077) = .308
Ace after one card is removed = 4/51 = .078
(.308)*(.078) = .024
4
3
2
1
Card
K (.077)
Q (.077)
J (.077)
10 (.077)
9 (.077)
8 (.077)
7 (.077)
6 (.077)
5 (.077)
4 (.077)
3 (.077)
2 (.077)
0
Ace (.077)
Frequency
5
Practice
• What is the probability of rolling a “1” using
a six sided dice?
• What is the probability of rolling either a
“1” or a “2” with a six sided dice?
• What is the probability of rolling two “1’s”
using two six sided dice?
Practice
• What is the probability of rolling a “1” using
a six sided dice?
1 / 6 = .166
• What is the probability of rolling either a
“1” or a “2” with a six sided dice?
• What is the probability of rolling two “1’s”
using two six sided dice?
Practice
• What is the probability of rolling a “1” using
a six sided dice?
1 / 6 = .166
• What is the probability of rolling either a
“1” or a “2” with a six sided dice?
(.166) + (.166) = .332
• What is the probability of rolling two “1’s”
using two six sided dice?
Practice
• What is the probability of rolling a “1” using
a six sided dice?
1 / 6 = .166
• What is the probability of rolling either a
“1” or a “2” with a six sided dice?
(.166) + (.166) = .332
• What is the probability of rolling two “1’s”
using two six sided dice?
(.166)(.166) = .028
Next step
• Is it possible to apply probabilities to a
normal distribution?
Theoretical Normal Curve
-3
-2
-1

1
2
3
Theoretical Normal Curve
Z-scores
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
We can use the theoretical normal distribution to
determine the probability of an event.
For example, do you know the probability of getting a Z
score of 0 or less?
.50
Z-scores
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
We can use the theoretical normal distribution to
determine the probability of an event.
For example, you know the probability of getting a Z
score of 0 or less.
.50
Z-scores
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
With the theoretical normal distribution we know the
probabilities associated with every z score! The
probability of getting a score between a 0 and a 1 is
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
What is the probability of getting a score of 1 or
higher?
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
These values are given in Table C on page 390
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
To use this table look for the Z score in column A
Column B is the area between that score and the
mean
Column B
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
To use this table look for the Z score in column A
Column C is the area beyond the Z score
Column C
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
The curve is symmetrical -- so the answer for a
positive Z score is the same for a negative Z
score
Column B
Column C
.3413
.3413
.1587
Z-scores
.1587
-3
-2
-1

1
2
3
-3
-2
-1
0
1
2
3
Practice
• What proportion of the normal distribution
is found in the following areas (hint: draw
out the answer)?
• Between mean and z = .56?
• Beyond z = 2.25?
• Between the mean and z = -1.45
Practice
• What proportion of the normal distribution
is found in the following areas (hint: draw
out the answer)?
• Between mean and z = .56?
.2123
• Beyond z = 2.25?
• Between the mean and z = -1.45
Practice
• What proportion of the normal distribution
is found in the following areas (hint: draw
out the answer)?
• Between mean and z = .56?
.2123
• Beyond z = 2.25?
.0122
• Between the mean and z = -1.45
Practice
• What proportion of the normal distribution
is found in the following areas (hint: draw
out the answer)?
• Between mean and z = .56?
.2123
• Beyond z = 2.25?
.0122
• Between the mean and z = -1.45
.4265
Practice
• What proportion of this class would have
received an A on the last test if I gave A’s
to anyone with a z score of 1.25 or higher?
• .1056
Note
• This is using a hypothetical distribution
• Due to chance, empirical distributions are not
always identical to theoretical distributions
• If you sampled an infinite number of times they
would be equal!
• The theoretical curve represents the “best
estimate” of how the events would actually
occur
Theoretical Distribution
5
Frequency
4
3
2
1
0
Ace
2
3
4
5
6
7
Card
8
9
10
J
Q
K
Empirical Distribution based on 52 draws
5
Frequency
4
3
2
1
0
Ace
2
3
4
5
6
7
Card
8
9
10
J
Q
K
Empirical Distribution based on 52 draws
5
Frequency
4
3
2
1
0
Ace
2
3
4
5
6
7
Card
8
9
10
J
Q
K
Empirical Distribution based on 52 draws
5
Frequency
4
3
2
1
0
Ace
2
3
4
5
6
7
Card
8
9
10
J
Q
K
Theoretical Normal Curve

Empirical Distribution
50
40
30
20
Count
10
0
1.13
1.63
1.38
BFISUR
2.13
1.88
2.63
2.38
3.13
2.88
3.63
3.38
4.13
3.88
4.63
4.38
4.88
Empirical Distribution
40
30
20
Count
10
0
1.60
2.20
2.00
BFIOPN
2.60
2.40
3.00
2.80
3.40
3.20
3.80
3.60
4.20
4.00
4.60
4.40
5.00
4.80
Empirical Distribution
40
30
20
Count
10
0
1.25
1.75
1.50
BFISTB
2.25
2.00
2.75
2.50
3.25
3.00
3.75
3.50
4.25
4.00
4.88
4.50
PROGRAM
http://www.jcu.edu/math/isep/Quincunx/Quin
cunx.html
Theoretical Normal Curve

Normality frequently occurs in many situations of psychology,
and other sciences
Practice
– #7.7
– #7.8