Sampling distribution of the mean and Central Limit Theorem

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Sampling distribution of the mean andCentral
Limit Theorem
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Data

• A small apartment building has 3 apartments.

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Find m and s

• Use the TI to obtain the values.
• The values are
• m
• s

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Form samples of size 2

• We need to form all samples of size 2, using
  replacement since the population is very small.
• Then we find the sample mean for each sample of 2
  apartments.

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Find m and s

• Use the TI to obtain the values.
• The values are
• m
• s

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ResultsMean of Sample means

• Mean of population equals mean of the sample
  means

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Results of Standard deviation of the sample means

• S.D. equals the population standard deviation
  divided by the square root of the sample size

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Example

• If m 40, s 10 and n 4, find
• If n 25, find

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Distribution of the sample means

• If the population is normally distributed, then
  the sample means will be normally distributed.
• If the population is not normally distributed,
  then the sample means will be normally
  distributed if the sample size is at least 30.

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Important Consequence

• If we take samples of size n from some
  population, under the previous conditions, then
  we can determine the probability of the sample
  means fulfilling some condition. We use

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Example 1

• The heights of kindergarten children are
  approximately normally distributed with a mean of
  39 and a standard deviation of 2. If one child
  is randomly selected, what is the probability
  that the child is taller than 41 inches?
• This is 1 child Not the Central Limit Theorem!

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Example 2

• Suppose we have a class of 30 kindergarten
  children. What is the probability that the mean
  height of these children exceeds 41 inches?
• This is the Central Limit Theorem as it is asking
  about the probability of a sample mean!

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Conclusion

• It is not unusual for one child, selected at
  random from a kindergarten class, to be taller
  than 41 inches.
• It is highly unlikely that the mean height for 30
  kindergarten students exceeds 41 inches.

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An analogy

• It would not be unusual for a student to get an A
  on a statistics test.
• It would be unusual if the class average for a
  statistics class was an A!

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Another Example

• The IQ scores of adults are normally distributed
  with a mean of 100 and a standard deviation of
  15.
• If one person is randomly selected, what is the
  probability that his IQ will be greater than 110?
• 2. If ten people are randomly selected, what is
  the probability that mean of those 10 peoples IQ
  will be greater than 110?

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Your Turn

• The weights of healthy adult female deer in
  December in Mesa Verde National Park is normally
  distributed with a mean of 63.0 kg and a standard
  deviation of 7.1 kg.
• If 1 such deer is randomly selected, what is the
  probability that it weighs less than 54 kg?
• If 20 deer are randomly selected, what is the
  probability that the sample mean is less than 20
  kg?