Transcript Lecture12

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.
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.
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.
Example
Find the probability x is at most 5 and at
least 2.
P(x)
0.1
0
0
2
4
6
P(2  x  5)
8
10
x
Example
Find the probability x is at most 5 and at
least 2.
P(x)
0.1
0
0
2
4
6
P(2  x  5)
8
10
x
Example
Find the probability x is at most 5 and at
least 2.
P(x)
0.1
0
0
2
4
6
P(2  x  5)
8
10
x
Example
Find the probability x is at most 5 and at
least 2.
P(x)
0.1
0
0
2
4
6
8
10
P(2  x  5)  (3)(0.1)  0.3
x
Normal Distributions
This is the most common observed
distribution of continuous random variables.
A normal distribution corresponds to bellshaped curves.
y 
e
( x   ) 2

/ 2 2
2
Normal Distributions
Shape of this curve is determined by µ and σ
– µ it’s centered, σ is how far it’s spread out.
Standard Normal Distribution
The Standard Normal Distribution is a
normal probability distribution that has a
mean of 0 and a standard deviation of 1.
  0,
1
In this way the formula giving the heights of
the normal curve is simplified greatly.
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.
z 
x

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
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.
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
Problems
Problems 5.3, 5.4, 5.12
Problems 5.22, 5.26, 5.28, 5.30, 5.36, 5.40,
5.48
Keys to success
Learn the standard normal table and how to
use it.
We will be using these tables through out
the course.
5.4 How do you know when a data
set is normal?
3 methods
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.
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:
– Within 2 sd:
– Within 3 sd:
68%
95%
99.7%
of the data.
of the data.
of the data.
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
IQR
 1.3
s
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?
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?
IQR 0.44

 1.33
s
0.33
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
IQR 0.44

 1.33
s
0.33
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
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
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