Sampling Distributions

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Chapter 6

• Sampling Distributions

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Those who jump off a bridge in Paris are in
Seine. A backward poet writes inverse. A
man's home is his castle, in a manor of speaking.
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Sampling

• The Need
• Get information about a population without
  checking the entire population
• Advantages
• Cost
• Time
• Accuracy (can be achieved with low cost)
• Destruction is sometimes involved checking all
  is not possible.
• Insert Excel Simulation here

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Distribution of Means
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Visual Mean of Means
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Distribution of Sample Means

• Many different sample means are possible
• The sample means cluster closer to the population
  mean than the population values do.
• The larger the sample, the closer they cluster
  around the population mean
• Therefore the likelihood of a single sample mean
  being close to the true mean is high

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Distribution of Sample Means

• When trying to use a sample to estimate a
  population mean, we know we wont get the exact
  value
• We want some way of managing the error so as to
  be as close as we need to be
• We can decide on a margin of error that we are
  willing to accept (polls typically 2 - 4).
• We cannot eliminate the possibility of getting a
  value outside that range, but we can keep it
  small by adjusting the sample size.

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How Close Can We Get?

• The variance of the sample mean is the population
  variance divided by n (sample size)
• Thus larger ns bring smaller variances
• Lets look at an example. In order to understand
  the process, we will assume we actually know the
  true mean and variance. Each of the following
  graphs is from a computer simulation of taking
  100 samples from a normal population with µ15
  and s3, but with different sample sizes.

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µ15, s3, Sample Size 1 Number observed in
14,16 30
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µ 15, s3, Sample Size 4 Number observed in
14,16 52
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µ 15, s3, Sample Size 9 Number observed in
14,16 74
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µ 15, s3, Sample Size 16 Number observed in
14,16 81
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µ 15, s3, Sample Size 25 Number observed in
14,16 90
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µ 15, s3, Sample Size 36 Number observed in
14,16 97
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Number in 14,16 vs Sample Size
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So What?

• In Real Life, we dont know the true mean and
  variance. We want to estimate them.
• Furthermore, we will only take one sample, which
  represents just one data point from the
  distributions we have illustrated.
• We will probably NEVER know where in the
  distribution that data point is coming from.
• Under these conditions, how can we provide an
  estimate that is trustworthy?
• Clearly, the sample size directly affects the
  likelihood that the sample mean will be close to
  the true mean.

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Which one would you like to pick from?

• The situation You have 100 balls in an urn
  (left). Each has an odd number on it, which may
  be from 7-25, but you dont know how many of each
  there are. You will draw one ball and record its
  number. If this number matches the mean of the
  distribution, your company will make lots of
  money and you will get a promotion. However, you
  have the opportunity, for a sizable fee, to trade
  in the urn for the one on the right. If you do
  so, and are wrong, you will be fired because of
  the excessive expense you incurred.

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Does the name Pavlov ring a bell? Reading while
sunbathing makes you well red. When two
egotists meet, it's an I for an I.
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• Notes
• 1. the sample mean.
• 2. the standard deviation of the sample
  means.
• 3. The theory involved with sampling
  distributions described in the remainder of this
  chapter requires random sampling.
• Random Sample A sample obtained in such a way
  that each possible sample of a fixed size n has
  an equal probability of being selected.
• (Example Every possible handful of size n has
  the same probability of being selected.)

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The Central Limit Theorem

• The most important idea in all of statistics.
• Describes the sampling distribution of the sample
  mean.
• Examples suggest the sample mean (and sample
  total) tend to be normally distributed.

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• Distribution of Sample Means
• If all possible random samples of a particular
  size n are taken from any population with a mean
  m and a standard deviation s, the distribution of
  sample means will
• 1. have a mean equal to m.
• 2. have a standard deviation equal to
• Further, if the sampled population has a normal
  distribution, then the sampling distribution of
  will also be normal for samples of all sizes.
• Central Limit Theorem
• The distribution of sample means will come closer
  to normal as the sample size increases.

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• Graphical Illustration of the Central Limit
  Theorem

Distribution of n 2
Original Population
Distribution of n 10
Distribution of n 30
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• Example Consider a normal population with m 50
  and s 15. Suppose a sample of size 9 is
  selected at random. Find
• 1.
• 2.
• Solution
• Since the original population is normal, the
  distribution of the sample mean is also (exactly)
  normal.

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• Example A report stated that the day-care cost
  per week in Boston is 109. Suppose we accept
  this as the true (population) mean cost per week,
  and also know that the standard deviation is 20.
• 1. Find the probability that a sample of 50
  day-care centers would show a mean cost of 105
  or less per week.
• 2. Suppose the sample of 50 day-care centers
  results in a sample mean of 120. Does this
  provide evidence to refute the claim that the
  true mean is 109?
• Solution
• The shape of the original distribution is
  unknown, but the sample size, n, is large. The
  CLT applies.
• The distribution of is approximately normal.

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• To investigate the claim, we need to examine how
  likely a sample mean of 120 is, if the claim is
  true.
• Consider how far out in the tail of the sample
  mean distribution the value 120 is found.
• Compute the tail probability.
• Since the tail probability is so small, this
  suggests the observation of 120 is very rare (if
  the mean cost is really 109).
• There is evidence to suggest the claim of m
  109 is wrong.

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In democracy your vote counts. In feudalism your
count votes. She was engaged to a boyfriend
with a wooden leg but broke it off. A chicken
crossing the road is poultry in motion.