Section 5.1 Review

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Section 5.1 Review
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The Binomial Setting
1) There are a fixed number n of observations
2) The n observations are independent
3) Each observation falls into just one of two
categories, which, for convenience, we call
success and failure.
4) The probability of a success, call it p, is
the same for each observation
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Binomial Distributions
The distribution of the count X of successes in
the binomial setting is called the binomial
distribution with parameters n and p.
The parameter n is the number of observations
The parameter p is is the probability of a
success on any one observation.
The possible values of X are the whole numbers
from 0 to n.
We say that X is B(n,p)
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Binomial Probability Tables
(Table C, page T-7 to T-10)
Consider the previous transistor problem using
the B(10,.1) distribution.
Q What is the probability that we will pull an
SRS from this distribution with exactly one bad
transistor?
A P(X 1)
0.3874
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Binomial Mean and Standard Deviation
Q If a count X is B(n,p), what are the mean ?X
and the standard deviation ?X ?
A If a count X has the B(n,p) distribution, then

?X np
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Sample Proportions
In statistical sampling, we often want to
estimate the proportion p of successes in a
population.
Our estimator is the sample proportion of
successes
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Mean and Standard Deviation of a Sample Proportion
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Normal Approximations for Counts and Proportions

• Draw an SRS of size n from a large population
  having
• population proportion p of successes.

• When n is large, the sampling distributions of
  these
• statistics are approximately normal.

Note Use this when np ? 10 and n(1-p) ? 10
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End O Review
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The Sampling Distribution of a Sample Mean
Recall
1) Counts and proportions are discrete random
variables that describe categorical data.
2) Measured data is usually described with
continuous random variables.
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The Sampling Distribution of a Sample Mean
Recall
3) Averages are less variable than individual
observations
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The Sampling Distribution of a Sample Mean
Recall
4) Averages are more normal than individual
observations
2 obs.
1 obs.
10 obs.
25 obs.
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Sample Mean
Q What is a sample mean?
A This is an estimate of the mean of the
underlying population
Q How do we find the sample mean?
A 1) Select an SRS of size n from a population
2) Measure a variable X on each individual in the
sample
3) Let each observation be labeled X1, X2, , Xn
4) The sample mean is the average of the Xis
Note If the sample is large, each Xi can be
thought of as an independent random
variable
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Mean and Standard Deviation of a Sample
Mean
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Mean and Standard Deviation of a Sample
Mean
Example
The height of a randomly chosen
Jedi varies according to the N(73, 2.8)
distribution.

If Yoda asked the height of an SRS
of 100 Jedi, what is the mean and height of this
sampling distribution?
73 inches
0.28
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Q What is the shape of the distribution?
A It depends on the population.

So, if the population distribution is
normal, then so is the distribution of the
sample mean.
This leads to the following
If a population has the N(?, ?) distribution,
then the sample mean has the N(
) distribution.
?,
In the previous example, the sample mean of the
heights of 100 Jedi has the N(73, 0.28)
distribution.
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• So, the sample mean from a normal distribution
  is normal.

• We can extend these ideas to the following

Any linear combination of independent normal
random variables is also normally distributed.

• In other words, if X and Y are independent
  normal random
• variables, then so is aX bY.

• This means the following are normal

2X 7Y, 3X - 78Y, OX 2Y, X - Y, etc.
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Example
Ben and Anakin are playing in the
local Jedi golf tournament. Bens golf game X has
the N(60, 4) distribution.
Anakins golf game Y has the N(76, 12)
distribution.
Q What is the probability that Anakin will score
lower than Ben in the tourney?
In other words, what is P(Y lt X) ?
A
Notice that Y - X is a linear combination of two
independent random normal variables, so Y - X is
normal.
What is the mean of the variable Y - X ?
?Y-X
?Y - ?X
76 - 60 16
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Example
Ben and Anakin are playing in the
local Jedi golf tournament. Bens golf game X has
the N(60, 4) distribution.
Anakins golf game Y has the N(76, 12)
distribution.
Q What is the probability that Anakin will score
lower than Ben in the tourney?
In other words, what is P(Y lt X) ?
A
Notice that Y - X is a linear combination of two
independent random normal variables, so Y - X is
normal.
What is the standard deviation of the variable Y
- X ?
?2Y-X
?2Y
?2X
122 42
160
?Y-X
12.65
So, Y - X has the N(16, 12.65) distribution.
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Example
Ben and Anakin are playing in the
local Jedi golf tournament. Bens golf game X has
the N(60, 4) distribution.
Anakins golf game Y has the N(76, 12)
distribution.
Q What is the probability that Anakin will score
lower than Ben in the tourney?
In other words, what is P( Y lt X) ?
So, Y - X has the N(16, 12.65) distribution.
P(Y lt X) P( Y - X lt 0)
P ( Z lt -1.26 )
.1038
So, Anakin will score lower about 10 of the time.
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Central Limit Theorem

• Draw an SRS of size n from any population with
  mean ?
• and finite standard deviation ?.When n is
  large, the
• sampling distribution of the sample mean is

Notes

• The distribution of a sum or average of many
  small random
• quantities is close to normal.

• This is true if the quantities are not
  independent or even
• if they have different distributions.

• How large does n have to be depends on the
  distribution.

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(Revisited)
How do we find the sampling distribution ?
1) Take repeated random samples of size n from a
population with mean ?
2) Find the sample mean for each sample
3) Collect all the sample means and display their
distribution.
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(Revisited)
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Homework
28, 31, 32, 37, 45