ECON 2300 LEC

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ECON 2300 LEC 8

• 10/05/06

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Outline

• Introduction - Sampling Distribution
• Sampling Distribution of
• Expected Value of
• Standard Deviation of
• Form of Sampling Distribution of
• Central Limit Theorem
• Practical value of the Sampling Distribution of
• Relationship Sample Size and Sampling
  Distribution of

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Introduction Sampling Distribution

• Sampling Distribution Set of values that we
  would obtain if we drew an infinite number of
  random samples from a given sample and calculated
  the statistic on each sample.
• All samples must be of the same size (n)
• Infinite samples not always possible
• Example Suppose that our population consists of
• only 3 numbers 2, 3 and 4. Our plan is to draw
• infinite number of samples of size n 2 and form
• a sampling distribution of the sample means.

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Introduction Sampling Distribution
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Sampling Distribution of x

• Process of Statistical Inference

Population with mean m ?
A simple random sample of n elements is
selected from the population.
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Sampling Distribution of x

• Example Director of personnel for Electronics
  Associates, Inc. (EAI), has been assigned the
  task of developing a profile of the companys
  2500 managers. The characteristics to be
  identified include the mean annual salary for the
  managers and the proportion of managers having
  completed the companys management training
  program.
• Using the population data, the population mean
  was obtained equal to 51,800 and population
  standard deviation equal to 4000 and population
  proportion equal to 0.6

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Sampling Distribution of x

• Using a sample of 30 EAI managers as shown in the
  table, the following results obtained
• Sample mean 51814
• Sample proportion 0.63
• Another sample yields
• Sample mean 52760
• Sample proportion 0.7
• If we repeat the same process over and over again
  and compute the statistical values, the resulting
  distribution sampling distribution

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Sampling Distribution of x
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Sampling Distribution of

• The sampling distribution of is the
  probability distribution of all possible values
  of the sample
• mean .
• Expected Value of
• E( ) ?
• where
• ? the population mean

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Sampling Distribution of

• Standard Deviation of
• Finite Population Infinite
  Population
• A finite population is treated as being
  infinite if n/N lt .05.
• is the finite correction
  factor.
• is referred to as the standard error of the
  mean.

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Sampling Distribution of

• If we use a large (n gt 30) simple random sample,
  the central limit theorem enables us to conclude
  that the sampling distribution of can be
  approximated by a normal probability
  distribution.
• When the simple random sample is small (n lt 30),
  the sampling distribution of can be
  considered normal only if we assume the
  population has a normal probability distribution.

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Sampling Distribution of

• Example (EAI managers)
• Population (N) 2500 managers (Finite)
• Standard deviation (s)4000
• Sample size (n)30
• n/N 30/2500.012, Finite population factor can
  be ignored,
• 4000/v30730.3

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Form of the Sampling Distribution of

• Population with normal distribution
• When the population has a normal distribution,
  the sampling
• distribution of
• Population without normal distribution
• When the population does not have a normal
  distribution, the Central Limit Theorem helps in
  identifying the shape of the sampling
  distribution of
• Central Limit Theorem
• In selecting simple random samples of size n from
  a population, the sampling distribution of the
  sample mean can be approximated by a normal
  distribution as the sample size becomes large.

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Example St. Andrews

• Sampling Distribution of for the SAT Scores

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Example St. Andrews

• Sampling Distribution of for the SAT Scores
• What is the probability that a simple random
  sample of 30 applicants will provide an estimate
  of the population mean SAT score that is within
  plus or minus 10 of the actual population mean ?
  ?
• In other words, what is the probability that
  will be between 980 and 1000?

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Example St. Andrews

• Sampling Distribution of for the SAT Scores

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Example St. Andrews

• Sampling Distribution of for the SAT Scores
• Using the standard normal probability table with
  
• z 10/14.6 .68, we have area (.2518)(2)
  .5036
• The probability is .5036 that the sample mean
  will be within /-10 of the actual population
  mean.

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Example St. Andrews

• Sampling Distribution of for the SAT Scores

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Relationship between sample size and the sampling
distribution of
Sampling distribution of
Sampling distribution of
With n100,
With n30,
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Relationship between sample size and the sampling
distribution of

• As sample size is increased, the standard error
  of the mean decreases.
• Larger sample size provides a higher probability
  that the sample mean is within a specified
  distance of the population mean.

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Sampling distribution of
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Sampling Distribution of x

• Sampling Distribution of x is the probability
  distribution of all possible values of the sample
  mean x
• Expected value of x