Download Class 4 Handout

Class 4
Normal Distributions
Sampling Distributions
Central Limit Theorem
Normal Random Variable
• Bell shaped curve
-3
-2
f ( x) 
-1
1
e
 2
0
1
1 x   
 

2  
2
3
2
  x
Computing Normal Probabilities
• We have computed probabilities for Z, a
standard normal. What is E(Z)?
Var(Z)?
• It turns out that if X is any normal random
variable with mean  and standard deviation
, then (X - )/ is a standard normal
random variate. As a result, we write
Z

X 

.
Computing Normal Probs. (cont.)
• Suppose that X has a normal distribution
with  = 5 and  = 3. Can you graph the
distribution of X?
• What is the P{X  11}?
X  5 11  5
P{ X  11}  P{

}
3
3
Computing Normal Probs (cont.)
• What is P{-1  X  9}?
1 5 X  5 9  5
 P{


}
3
3
3
Using EXCEL
• Select the Function Wizard (fx)
statistical/normdist.
• The syntax of this function looks like
normdist(x, , , true or false).
• If the fourth argument is true, this will
return P{X  x} where X is a normal(, ).
Example
• How can you interpret the computation of
Z?
• The lifetime of a tire has a normal
distribution with a mean of 40,000 miles
and a standard deviation of 3,000 miles. It
is desired to set a warranty on these tires
such that 10% of the tires fall under
warranty. What is the required value (in
miles)?
Example (cont.)
• How many standard deviations do we have
to go out on a (any) normal distribution to
cut off 10%?
• Therefore, if w is the warranty limit, we
have:
Using EXCEL
• The function norminv(prob, , ) will
return the value on a normal(, )
distribution that cuts off prob to the left.
• Try norminv(.1, 40000, 3000).
Summary (So far)
• Describe Data Graphically and Numerically
• Populations vs. Samples
• Further description of populations--Random
Variables
• Discrete
• Continuous
Now we will return to sampling and apply what
we have learned.
Sampling
• Reasons for sampling as opposed to taking a census
•
•
•
•
Cost
Speed
Analysis
Feasibility
• Types
• Nonrandom
• Random
Judgment or Convenience
» Simple Random Sample: A sample where all samples of
size n have the same chance of being chosen.
» Systematic
» Stratified
» Cluster
Sampling Distributions
• Basic idea: Imagine all simple random
samples of size n that can be drawn from a
population. Each sample has its own
characteristics (such as a sample mean). We
might wonder about the likelihood of seeing
a particular characteristic in our sample.
This is the idea behind a sampling
distribution.
Example
• Infinite population:
X
1
2
4
p(x)
0.2
0.2
0.6
• For future reference:
 = 1(.2) + 2(.2) + 4(.6) = 3
2 = (1-3)2(.2) + (2-3)2(.2) + (4-3)2(.6)
= 1.6
Distribution of X
0.6
0.6
0.5
Probabilities
0.4
0.3
0.2
0.2
0.2
0.1
0
0
1
2
3
4
Distribution of Sample Mean (n = 2)
0.6
0.5
0.4
Prob
0.36
0.3
0.24
0.24
0.2
0.1
0.08
0.04
0.04
0
0
1
1.5
2
2.5
3
3.5
4
Distribution of Sample Mean (n = 3)
0.25
0.216
0.216
0.216
0.2
0.144
Prob
0.15
0.1
0.08
0.072
0.05
0.024
0.024
0.008
0
0
1.000
1.333
1.667
2.000
2.333
2.667
3.000
3.333
3.667
4.000
Distribution of Mean (n = 4)
0.2000
0.1800
0.1600
0.1400
Prob
0.1200
0.1000
0.0800
0.0600
0.0400
0.0200
0.0000
1.000
1.250
1.500
1.750
2.000
2.250
2.500
2.750
3.000
3.250
3.500
3.750
4.000
4.
3.
3.
3.
3.
3.
3.
3.
2.
2.
2.
2.
2.
2.
2.
1.
1.
1.
1.
1.
1.
1.
0
00
7
85
4
71
1
57
9
42
6
28
3
14
0
00
7
85
4
71
1
57
9
42
6
28
3
14
0
00
7
85
4
71
1
57
9
42
6
28
3
14
0
00
Prob
Distribution of Sample Mean (n = 7)
0.1400
0.1200
0.1000
0.0800
0.0600
0.0400
0.0200
0.0000
Homework
• For the following population:
X
0
1
2
p(x)
0.5
0.3
0.2
• Compute  and .
• Generate the sampling distribution of X for
a sample of size n = 3.
The Central Limit Theorem
• Let X be computed by taking a simple
random sample of size n from a population
with mean  and standard deviation .
Then for large n, the distribution of X will
be approximately normal. Large n means:
• n  1 when sampling from a normal distribution,
• n  30 when sampling from any distribution.
As always when sampling from an infinite population or a population
of size N where N>>n,
E ( X )   , and Var( X ) 
2
n
.
A Quick Note

Var ( X )   
n

 
2
X
X
n
2
Using the CLT
• Incomes in a community are normally
distributed with a mean of $25,000 and
standard deviation of $8,000. If we take a
sample of size 4, what is the probability that
the average income in the sample is greater
than $29,000?
Income Example
  25,000

  8,000  
n4


 X  25,000
X 
n
 8000
4
 4000
 X   X 29,000  25,000 
P{ X  29,000}  P 


4,000
 X

 P{Z  1}
 .1587
Income Example
• What is the probability that a single
income selected will fall above
$29,000?
X   29,000  25,000
P{ X  29,000}  P{

}

8,000
 P{Z  0.5}
 0.3085
• What proportion of the population will
fall above $29,000?
Using the CLT
• A company produces lids for tin cans. The
lids are supposed to be 4 inches in diameter.
The standard deviation of tin can lids is .012
inches. Because a worker suggested that
the machine is in need of adjustment, the
foreman has taken a sample of 100 lids and
found that x  4.003 inches. Should they
shut down production to make the
adjustment?