Sampling Distributions

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Sampling Distributions

• Chapter 9

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9.1 Introduction

• In real life calculating parameters of
  populations is prohibitive because populations
  are very large.
• Rather than investigating the whole population,
  we take a sample, calculate a statistic related
  to the parameter of interest, and make an
  inference.
• The sampling distribution of the statistic is the
  tool that tells us how close is the statistic to
  the parameter.

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9.2 Sampling Distribution of the Mean

• An example
• A die is thrown infinitely many times. Let X
  represent the number of spots showing on any
  throw.
• The probability distribution of X is

E(X) 1(1/6) 2(1/6) 3(1/6) .
3.5 V(X) (1-3.5)2(1/6) (2-3.5)2(1/6)
.
2.92
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Throwing a die twice sample mean

• Suppose we want to estimate m from the mean of
  a sample of size n 2.
• What is the distribution of ?

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Throwing a die twice sample mean
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The distribution of when n 2
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Sampling Distribution of the Mean
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Sampling Distribution of the Mean
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Sampling Distribution of the Mean
Demonstration The variance of the sample mean is
smaller than the variance of the population.
Mean 1.5
Mean 2.5
Mean 2.
Population
1.5
2.5
2
1
2
3
1.5
2.5
2
1.5
2.5
2
1.5
2.5
2
1.5
2.5
2
Compare the variability of the population to the
variability of the sample mean.
1.5
2.5
1.5
2.5
2
1.5
2.5
2
1.5
2.5
2
1.5
2.5
1.5
2.5
2
1.5
2.5
2
Let us take samples of two observations
1.5
2.5
2
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Sampling Distribution of the Mean
Also, Expected value of the population (1 2
3)/3 2
Expected value of the sample mean (1.5 2
2.5)/3 2
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The Central Limit Theorem

• If a random sample is drawn from any population,
  the sampling distribution of the sample mean is
  approximately normal for a sufficiently large
  sample size.
• The larger the sample size, the more closely the
  sampling distribution of will resemble a
  normal distribution.

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Sampling Distribution of the Sample Mean
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Sampling Distribution of the Sample Mean

• Example 9.1
• The amount of soda pop in each bottle is normally
  distributed with a mean of 32.2 ounces and a
  standard deviation of .3 ounces.
• Find the probability that a bottle bought by a
  customer will contain more than 32 ounces.
• Solution
• The random variable X is the amount of soda in a
  bottle.

0.7486
m 32.2
x 32
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Sampling Distribution of the Sample Mean

• Find the probability that a carton of four
  bottles will have a mean of more than 32 ounces
  of soda per bottle.
• Solution
• Define the random variable as the mean amount of
  soda per bottle.

0.9082
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Sampling Distribution of the Sample Mean

• Example 9.2
• Deans claim The average weekly income of B.B.A
  graduates one year after graduation is 600.
• Suppose the distribution of weekly income has a
  standard deviation of 100. What is the
  probability that 25 randomly selected graduates
  have an average weekly income of less than 550?
• Solution

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Sampling Distribution of the Sample Mean

• Example 9.2 continued
• If a random sample of 25 graduates actually had
  an average weekly income of 550, what would you
  conclude about the validity of the claim that the
  average weekly income is 600?
• Solution
• With m 600 the probability of observing a
  sample mean as low as 550 is very small (0.0062).
  The claim that the mean weekly income is 600 is
  probably unjustified.
• It will be more reasonable to assume that m is
  smaller than 600, because then a sample mean of
  550 becomes more probable.

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9.3 Sampling Distribution of a Proportion

• The parameter of interest for nominal data is the
  proportion of times a particular outcome
  (success) occurs.
• To estimate the population proportion p we use
  the sample proportion.

The estimate of p
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9.3 Sampling Distribution of a Proportion

• Since X is binomial, probabilities about can
  be calculated from the binomial distribution.
• Yet, for inference about we prefer to use
  normal approximation to the binomial.

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Normal approximation to the Binomial

• Normal approximation to the binomial works best
  when
• the number of experiments (sample size) is large,
  and
• the probability of success, p, is close to 0.5.
• For the approximation to provide good results two
  conditions should be met
• np 5 n(1 - p) 5

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Normal approximation to the Binomial
Example Approximate the binomial probability
P(x10) when n 20 and p .5 The parameters of
the normal distribution used to approximate the
binomial are m np s2 np(1 - p)
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Normal approximation to the Binomial
Let us build a normal distribution to approximate
the binomial P(X 10).
m np 20(.5) 10 s2 np(1 - p)
20(.5)(1 - .5) 5 s 51/2 2.24
P(XBinomial 10)
.176
10
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Normal approximation to the Binomial

• More examples of normal approximation to the
  binomial

P(Ylt 4.5)
P(X 4) _at_
4
P(X ³14) _at_
P(Y gt 13.5)
14
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Approximate Sampling Distribution of a Sample
Proportion

• From the laws of expected value and variance, it
  can be shown that E( ) p and V( )
• p(1-p)/n
• If both np gt 5 and np(1-p) gt 5, then
• Z is approximately standard normally distributed.

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• Example 9.3
• A state representative received 52 of the votes
  in the last election.
• One year later the representative wanted to study
  his popularity.
• If his popularity has not changed, what is the
  probability that more than half of a sample of
  300 voters would vote for him?

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• Example 9.3
• Solution
• The number of respondents who prefer the
  representative is binomial with n 300 and p
  .52. Thus, np 300(.52) 156 andn(1-p)
  300(1-.52) 144 (both greater than 5)