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Review
A random variable where X can take on a range of values, not just particular ones.
Examples:
Heights
Distance a golfer hits the ball with their driver
Time to run 100 meters
Electricity usage of a home.
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Review
There are two types of continuous distributions we discuss now: uniform and normal distributions.
A density curve is the graph of a continuous probability distribution.
1) This curve always positive (or 0)
2) The area under the curve is 1.
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Review For a density curve depicting the
probability distribution of a continuous random variable, – the total area under the curve is 1, – there is a direct correspondence between area
and probability.– Only the probability of an event occurring in
some interval can be evaluated. – The probability that a continuous random
variable takes on any particular value is zero.
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Example
Find the probability x is at most 5 and at least 2.
x
P(x)
0 2 4 6 8 10
0.1
0
)52(P x
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Example
Find the probability x is at most 5 and at least 2.
x
P(x)
0 2 4 6 8 10
0.1
0
)52(P x
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Example
Find the probability x is at most 5 and at least 2.
x
P(x)
0 2 4 6 8 10
0.1
0
)52(P x
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Example
Find the probability x is at most 5 and at least 2.
x
P(x)
0 2 4 6 8 10
0.1
0
3.0)1.0)(3()52(P x
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Normal Distributions
This is the most common observed distribution of continuous random variables. A normal distribution corresponds to bell-shaped curves.
2
/ 22 2)(
xe
y
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Normal Distributions
Shape of this curve is determined by µ and σ – µ it’s centered, σ is how far it’s spread out.
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Standard Normal Distribution
The Standard Normal Distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.
In this way the formula giving the heights of the normal curve is simplified greatly.
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Z-score
We represent a standard normal variable with a z instead of an x.
Convert any normal distribution to a standard normal distribution by using the
z-score.
x
z
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Standard Normal Probabilities
P(0 z 1) = 0.3413
This can found in a table in the back of the text (Table IV). The table only gives the areas under the curve to the right between 0 and z. To find other intervals requires some tricks
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Finding Probabilities when given z-scores.
For a given z-score, the probability can be found in a table in the back of the text (Table IV), also see inside front cover.
Note: The table only gives the areas under the curve to the right between 0 and z. To find other intervals requires some tricks.
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Examples
Use the tables in the back of the book to find the following.
a) P(0 z 2.43) = 0.4925
b) P(-2.43 z 0) = 0.4925
c) P(1.20 z 2.30) =0.4893 - 0.3849=0.1044
d) P(-1.50 z 2.4) = 0.4918 + 0.4332 = 0.925
e) P( z 1.8) = 0.4641 + 0.5 = 0.9641
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Problems
Problems 5.3, 5.4, 5.12
Problems 5.22, 5.26, 5.28, 5.30, 5.36, 5.40, 5.48
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Keys to success
Learn the standard normal table and how to use it.
We will be using these tables through out the course.
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5.4 How do you know when a data set is normal?
3 methods
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5.4 How do you know when a data set is normal?
Method 1:
• A data set is approximately normal if it is symmetric and bell-shaped.
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5.4 How do you know when a data set is normal?
Method 2:
• A set of data is approximately normal if the data set satisfies the empirical rule:
– Within 1 sd: 68% of the data.– Within 2 sd: 95% of the data.– Within 3 sd: 99.7% of the data.
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5.4 How do you know when a data set is normal?
Method 3:
• Find IQR and standard deviation. If the data is approximately normal, then
3.1s
IQR
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Example
You are given a data set and determine that: 1. IQR=0.44 and
2. s=0.33
Would you suspect this data is normally distributed?
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Example
You are given a data set and determine that: 1. IQR=0.44 and
2. s=0.33
Would you suspect this data is normally distributed?
33.133.0
44.0
s
IQR
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Example
You are given a data set and determine that: 1. IQR=0.44 and
2. s=0.33
Would you suspect this data is normally distributed? YES
33.133.0
44.0
s
IQR
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Problems
Problems 5.3, 5.4, 5.12
Problems 5.22, 5.26, 5.28, 5.30, 5.36, 5.40, 5.48
Problem 5.54
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25
Homework
• Review Chapter 5.1-5.4
• Read Chapters 6.1-6.3 for next week
• Midterm on Thursday, – 7:00-8:30 in PS 1072– Covers chapters 1-4
• Quiz during class on Tuesday
• Next class - optional tutorial