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Unit 5
Probability Distribution
MM207
JoEllen Green, MS
Normal Distribution Characteristics
• Most Statistical test are based on
the Normal Curve.
You have already hear about the normal
curve in school when some students would
ask their instructor, “Are you grading on a
curve?”
What does grading on a curve mean?
0.2
0.15
0.1
0.05
F
0
-∞
-9
-8
-7
-6
-5
-4
-3
C
D
-2
-1
0
B
1
A
2
3
P(x)
4
5
6
7
8
P(x)
9
∞
Characteristics of a
Normal Distribution
It is a continuous distribution.
0.2
0.15
0.1
0.05
0
-∞
-9
-8
-7
-6
-5
-4
-3
-2
-1
P(x)
0
1
2
3
4
5
6
7
8
P(x)
9
∞
Normal Curve Characteristics
It is symmetric on both sides of the mean.
0.2
0.15
0.1
0.05
0
-∞
-9
-8
-7
-6
-5
-4
-3
-2
-1
P(x)
0
1
2
3
4
5
6
7
8
P(x)
9
∞
Normal Curve Characteristics
Values range from -∞ to ∞
0.2
0.15
0.1
0.05
0
-∞
-9
-8
-7
-6
-5
-4
-3
-2
-1
P(x)
0
1
2
3
4
5
6
7
8
P(x)
9
∞
Normal Curve Characteristics
Mean = Median = Mode
0.2
0.15
0.1
0.05
0
-∞
-9
-8
-7
-6
-5
-4
-3
-2
-1
P(x)
0
1
2
3
4
5
6
7
8
P(x)
9
∞
Normal Curve Characteristics
Follows the Empirical Rule
34%
13.5%
0.015%
2.35%
34%
13.5%
2.35%
0.015%
Standard Normal Curve
• Mean = µ = 0
• Standard Deviation= σ = 1
• We have tables to find the
probability for this distribution
Finding Probabilities with
Standard Normal
• Turn to page A16 and A17 in your text
book or take the chart that folds out in
the back of the book.
• We will use Table 4.
Finding the Probabilities
• Find the probability that Z < -1.35.
• Word Phrases: to the left of -1.35
less than -1.35
• P(Z < -1.35)
• When I show you how to find the P(Z < -1.35),
You can find the P(Z < -1.96).
To Calculate P( Z> 1.25)
• What is the P(Z < 1.25) = 0.8944
• Subtract it from 1?
• What is your answer?
1 – 0.8944
0.1056
Finding other Probabilities
• Find P(Z > -1.35)
Other phrases:
more than -1.35
greater than -1.35
Finding other Probabilities
• P(Z > -1.96) =
• P(Z < 1.23) =
• P(Z > 1.65) =
Computing P( -1.50 < Z < 1.25) = ?
A. What is P(Z < 1.25) = ?
B. What is the P(Z < -1.50)?
C. Subtract B from A?
Finding the Probabilities
• P( -1.35 < Z < 1.23) =
• P( 0.53 < Z < 1.88 )=
• P(-0.15 < Z < 2.53) =
P( Z< -1.20 or Z > 1.30)
• Do the ranges overlap?
Computing P( Z< -1.20 or Z > 1.30)
• Compute P( Z < -1.20)
• Compute P( Z > 1.30)
• Add them together
Compute the following probabilities
• P( Z < 0 or Z > 2.33) =
• P( Z < -2.00 or Z > 1.30)
How to compute a Z score
• If your mean is not 0 and the standard
deviation is not 1, can you use the tables to
compute the probabilities?
Z Score
Find the Probability
• The Mean µ = 86 and standard deviation σ = 5.
• Find the probability P(X < 80)
•
•
•
•
What is the Value?
What is the Mean?
What is the Standard Deviation?
Compute Z
Finish the Problem
• P(X < 80) = P(Z < -1.20) =
• Find P(X < 100)
Calculating Probabilities
• Find P( 70 < X < 80)
A)
Find Z Score for 70
B)
Find the Probability for
Find the Z Score for 80
c)
Find the Probability for
Subtract A from B
Calculating Probabilities
• Mean µ = 86 and Standard Deviation σ = 5.
• P(x < 75)
• P( 85 < X < 95)
Page 257 #14
The lengths of Atlantic croaker fish are normally
distributed, with a mean of 10 inches and a standard
deviation of 2 inches. An Atlantic croaker fish is
randomly selected.
a)
Find the probability that his height is less than 66
inches?
Central Limit Theorem
What if the distribution is not normal?
We need only gather a sample of size, n > 30 or more.
We can assume the sample is normal with mean µ and
standard deviation σ/√n
Find the Probability
Page 279 Problem #13
For a sample of n=36, find the probability of a sample
mean being less than 12.2 if µ = 12 and σ = 0.95.
What is the value?
What is the mean?
What is the standard deviation?
Compute the Z score?
What is the probability?
Find the Probability
Page 279 Problem #14
For a sample of n=100, find the probability of a
sample mean being greater than 12.2 if µ = 12 and σ =
0.95.
What is the value?
What is the mean?
What is the standard deviation?
Compute the Z score?
What is the probability?
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