Sampling Distributions

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Sampling Distributions

• A review by Hieu Nguyen(03/27/06)

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Parameter vs Statistic

• A parameter is a description for the entire
  population.
• ExampleA parameter for the US population is the
  proportion of all people who support President
  Bushs nomination of Samuel Alito to the Supreme
  Court.
• p.74

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Parameter vs Statistic

• A statistic is a description of a sample taken
  from the population. It is only an estimate of
  the population parameter.
• ExampleIn a poll of 1001 Americans, 73 of
  those surveyed supported Alitos nomination.
• p-hat.73

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Bias

• The bias of a statistic is a measure of its
  difference from the population parameter.
• A statistic is unbiased if it exactly equals the
  population parameter.
• ExampleThe poll would have been unbiased if 74
  of those surveyed approved of Alitos nomination.
• p-hat.74p

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Sampling Variability

• Samples naturally have varying results. The mean
  or sample proportion of one sample may be
  different from that of another.
• In the poll mentioned before p-hat.73.
• A repetition of the same poll may have
  p-hat.75.

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Central Limit Theorem (CLT)

• Populations that are wildly skewed may cause
  samples to vary a great deal.
• However, the CLT states that these samples tend
  to have a sample proportion (or mean) that is
  close to the population parameter.
• The CLT is very similar to the law of large
  numbers.

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CLT Example

• Imagine that many polls of 1001 Americans are
  done to find the proportion of those who
  supported Alitos nomination.
• Although the poll results vary, more samples have
  a mean that is close to the population parameter
  µ.74.

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CLT Example

• Plot the mean of all samples to see the effects
  of the CLT. Notice how there are more sample
  means near the population parameter µ.74.

• This histogram is actually a sampling
  distribution

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Sampling Distributions Definition

• Textbook definitionA sampling distribution is
  the distribution of values taken by the statistic
  in all possible samples of the same size from the
  same population.
• In other words, a sampling distribution is a
  histogram of the statistics from samples of the
  same size of a population.

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Two Most Common Types of Sampling Distributions

• Sample Proportion Distribution
• Distribution of the sample proportions of samples
  from a population
• Sample Mean Distribution
• Distribution of the sample means of samples from
  a population
• For both types, the ideal shape is a normal
  distribution

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Sampling Distributions Conditions

• Before assuming that a sampling distribution is
  normal, check the following conditions
• Plausible Independence
• Randomness
• Each sample is less than 10 of the population

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Sampling Distributions As Normal Distributions

• When all conditions met, the sampling
  distribution can be considered a normal
  distribution with a center and a spread.
• NoteWith sample proportion distributions,
  another condition must be meet
• Success-failure conditon there must be at least
  10 success and 10 failures according to the
  population parameter and sample size

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Sampling Distributions As Normal Distributions
Equations

• Sample Proportion Distribution
• p population proportion (given)

• Sample Mean Distribution
• µ population mean (given)
• s population standard deviation (given)

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Sampling Distributions As Normal Distributions
Note

• NoteIf any of the parameters are unknown, use
  the statistics from a sample to approximate it.

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Using Sampling Distributions

• Sampling Distributions can estimate the
  probability of getting a certain statistic in a
  random sample.
• Use z-scores or the NormalCDF function in the
  TI-83/84.

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Using Sampling Distributions Z-Scores w/ Example

• Use the z-score table to find appropriate
  probabilities
• ExampleFind the probability that a poll of
  Americans that support Alitos nomination will
  return a sample proportion of .72.

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Using Sampling Distributions NormalCDF Function
w/ Example

• The syntax for the NormalCDF function is
• NormalCDF(lower limit, upper limit, µ, s)
• ExampleFind the probability that a sample of
  size 25 will have a mean of 5 given that the
  population has a mean of 7 and a standard
  deviation of 3.

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Sampling Distribution for Two Populations

• Use a difference sampling distribution if the
  question presents 2 different populations.

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Sampling Distribution for Two Populations
Example

• (adapted from AP Statistics Chapter 9
  Sampling Distribution Multiple Choice Questions
• Medium oranges have a mean weight of 14oz and a
  standard deviation of 2oz. Large oranges have a
  mean weight of 18oz and a standard deviation of
  3oz. Find the probability of finding a medium
  orange that weights more than a large orange.

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Example Problem

• (adapted from DeVeau Sampling Distribution Models
  Exercise 42)
• Ayrshire cows average 47 pounds if milk a day,
  with a standard deviation of 6 pounds. For Jersey
  cows, the mean daily production is 43 pounds,
  with a standard deviation of 5 pounds. Assume
  that Normal models describe milk production for
  these breeds.
• A) We select an Ayrshire at random. Whats the
  probability that she averages more than 50 pounds
  of milk a day?
• B) Whats the probability that a randomly
  selected Ayrshire gives more milk than a randomly
  selected Jersey?
• C) A farmer has 20 Jerseys. Whats the
  probability that the average production for this
  small herd exceeds 45 pounds of milk a day?
• D) A neighboring farmer has 10 Ayrshires. Whats
  the probability that his herd average is at least
  5 pounds higher than the average for the Jersey
  herd?

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Example Problem Solution

• First, check the assumptions
• Independent samples
• Randomness
• Sample represents less than 10 of population

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Example Problem Solution

• A) Use the normal model to estimate the
  appropriate probability.

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Example Problem Solution

• B) Create a normal model for the difference
  between Ayrshires and Jerseys. Use the model to
  estimate the appropriate probability.

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Example Problem Solution

• C) Create a sampling distribution model for which
  n20 Jerseys. Use the model to estimate the
  appropriate probability.

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Example Problem Solution

• D) First create a sampling distribution model for
  10 random Ayrshires and 20 random Jerseys. Then
  create a normal model for the difference between
  the 10 Ayrshires and 20 Jerseys.