Download Sampling Distributions Central Limit Theorem - RIT

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Sampling Distributions
Central Limit Theorem
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Sampling Distribution
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Distribution of a statistic
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Recall
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Population : Collection of all data on a characteristic
Parameter : Measure of a population
Sample
Statistic
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e.g. mean, s.d, proportion, median, etc., of a population
notation: µ,σ,π, etc.
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: Any subset of a population
: Measure of a population
e.g. mean, s.d., proportion, median of sample data
notation:
x , s, p
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Objective:
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Learn properties of statistics (sample measures)
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In particular, properties of sample mean and
sample proportion
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What are the average and s.d. of the sample mean?
What is the distribution of the sample mean x ?
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What are the average and s.d. of sample proportion?
What is the distribution of the sample proportion, p?
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POPULATION (X)
MEAN : µ
S.D. : σ
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Choose random, independent samples each of size n.
1: x11, x 12, x13, ..., x1n
„Sample mean
1
„Sample
x
„Sample
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Sample 2 x21, x22, x23,...., x2n
„Sample
mean =
x2
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3: x31, x 32, x33, ..., x3n
mean
, etc.
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x
„Sample
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So, we could have many sample means.
Sample mean is a random variable
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A statistic
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What can we say about this?
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1. On the average, the sample means
must equal the population mean
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2. The sample means are closer together
than the population values themselves.
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Meaning:
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Average of sample means= population mean
µX = µ
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Variance of the sample means is smaller than
the variance of the population (σ2 )
How much smaller?
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Variance of sample means =
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σ 2x =
6
σ2
n
; so, σ X =
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population variance
sample size
σ
n
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Important Observations
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There are three “averages” we are talking about:
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1. Population mean = µ Å This is a constant
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2. Sample mean =
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3. Mean of Sample means
x
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Å This is a random variable
µX
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Å A constant
Similarly, there are two variances:
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1. Variance of the population= σ Å A constant
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2. Variance of Sample means=
σ X2
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Å A constant
You need to use the right notation and interpret
each quantity appropriately.
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µx = µ;
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So, we have the important relationship:
σ 2x =
σ
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2
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n
Note that the bigger the sample size, the less
variable are the values of sample means.
The above are true for means of samples
from any population and for samples of any
size.
What about the distribution of the sample
means, x ? Recall that x is a r.v.
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If population is normal then
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~ Normal
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x
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for large samples
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How large?
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x
If population is not normal or unknown, then
≈ Normal (approximately normal)
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Sample size at least 30.
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CENTRAL LIMIT THEOREM
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Applies when population is non-normal or when
the population distribution is unknown.
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Central Limit Theorem
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The random variable is the number of units a component
sold by a company. The probability distribution of X is
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given as:
U n its s o ld
1000
3000
5000
10000
P ro b a b ility
0 .1
0 .3
0 .4
0 .2
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We can compute the mean µ = 5000 units and the
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standard deviation is σ = 2792.8 units
Suppose we look at a sample of 16 sales and do this, say,
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100 times. We can easily mimic (simulate) this
experiment in Minitab.
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First we have to enter the probability
distribution in our worksheet.
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Column 1 has the values
1000,3000,5000.10000
Column 2 has the probabilities
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Now we tell Minitab to generate 100
samples each of size 16 from our discrete
distribution:
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Take a look at some of the samples…
Row
1 Sample 1
2 Sample 2
C3
10000
3000
C4
C5
C6
C7
C8
C9
C10
5000 5000 3000 3000 5000 1000 10000 ….
1000 3000 3000 3000 10000 5000 10000 …..
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Now, compute the means of each of the 100
samples store them in column c19
Look at the histogram of sample means in
C19 compare with the histogram of the
“population distribution”.
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0.4
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prob
Frequency
0.3
0.2
0.1
10
0
0
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
3500 4000 4500 5000 5500 6000 6500 7000
units
sam16
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Let us repeat the process by taking samples
of size 64, 81. Below you see the graphs of
the histograms of the sample averages for
all three cases.
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Frequency
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10
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3500 4000 4500 5000 5500 6000 6500 7000
sam16
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Frequency
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10
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0
4250 4500 4750 5000 5250 5500 5750 6000
sam64
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Frequency
30
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10
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0
4400 4600 4800 5000 5200 5400 5600 5800
sam81
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Note that:
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In all the three examples, the middle value -- the mean
of the sample means -- is about 5000 (the mean of the
population)
„ this is what the property
µ x = µ says
As the sample size increases, the values of the sample
mean get closer.
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this is what the property
σx =σ / n
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says.
The distribution gets more symmetrical.
(I.e.) As the sample size gets larger, the distribution
of the sample averages looks like that of a normal
distribution.
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this is what CLT says.
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Central Limit Theorem for Proportions
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There is nothing special about averages
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Binomial Population
Success / Failure
π= P(Success)
π= Proportion of successes
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Sample of size n : x1,x2,... , xn
Success / failure
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Examples
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Each component is either good or defective
π = Proportion of good items
in the population (Shipment)
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Shipment of components
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process dependent
may or may not be known
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Random sample of 250 components
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Each component is either good or bad
p = proportion of good components in the
sample
Changes from one sample to another sample
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Population:
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p = proportion (fraction) of voters in the
sample who favor the candidate
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Value of p will change from one sample to
another
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Random sample of 1057 voters
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Favorable or unfavorable opinion about a
candidate, say in our county.
π = proportion of voters who favor the
candidate
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Random
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• p = proportion of successes in the sample
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• Sample proportion
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• Statistic; Random
• What are the properties of the statistic p ?
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• (1) Average of sample proportion is the population proportion:
µp = π
• (2) S.d. of sample proportion p:
• CLT => p ≈ Normal
σp =
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π (1 − π )
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n
• Combine with (1) and (2) to get
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p≈
(
N π,
π (1 − π )
n
)
for large samples.
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Things to remember...
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Use appropriate notation to identify the
given information
Define the random variables
Write the question in terms of the r.v.
Keep the information about the population
and the information about the sample
separate.
Connect your steps.
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