Chapter 18 Sampling Distribution Models

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Chapter 18Sampling Distribution Models

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Drawing Conclusions- Inference

• Our whole purpose in learning stat so far has
  been to be able to draw conclusions about
  populations (which we cant measure) using
  information from samples.
• Need to know- How often would my method give a
  correct answer, if used many times?
• Inference is best when data is from
• Random Sample
• Randomized Comparative Experiment

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Sampling Distribution

• With our Reeses Pieces, we saw what is called a
  sampling distribution.
• Sample proportions (phat) will vary sample to
  sample
• Sampling Distribution- distribution of values
  taken on by a statistic in ALL possible samples
  of same size (n) from same population.
• Ideal pattern from looking at infinite number of
  samples

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Sampling Distributions

• With the Reeses, we saw a distribution that was
• Symmetric
• Unimodal
• Approximately Normal
• Center of distribution close to true proportion
  of population.

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Describing Sampling Distributions

• What do we need to describe a Normal
  distribution?
• m and s
• What should m be for our Reeses pieces samping
  distribution?
• Equal to actual population proportion, 0.45.
• What should s be?
• Interesting property for proportions- standard
  deviation is related to p.

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Sampling Dist for Proportions

• p proportion of successes
• 1-p proportion of failures (not p)
• Let q 1-p
• Then we can write the standard deviation of the
  sampling distribution of phat as
• So, we can describe the sampling distribution for
  proportions as (this is the important part!)

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Conditions and Assumptions

• In order to use the Normal model, we need to make
  the following assumptions/conditions
• Assumptions
• Sampled values are independent
• Sample size is large enough
• Conditions
• 10 condition- sample is no more than 10 of
  population
• n is large enough to have at least 10 successes
  and 10 failures (np gt 10 and nq gt10)

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Using the sampling dist

• Once we have the model for the sampling
  distribution, the problems turns back into one of
  z-scores and areas under the curve.

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Example 10 Contacts

• Assume that 30 of students at a university wear
  contact lenses. We randomly pick 100 students.
  What is the appropriate model for the
  distribution of p_hat?
• Check assumptions/conditions
• Assume people are SRS, 100 lt10 of pop.
• np 1000.30 30gt10
• nq100.70 70 gt 10
• Whats the appropriate model?
• Mean?
• SD?
• N(0.30, )
• N(0.3,0.0458)

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Contact example

• What is the probability that more than 1/3 of the
  sample wears contacts?
• In other words, what proportion of all possible
  samples will have more than 33 of the 100
  students wear contacts?
• (draw picture)
• How can we solve this?
• Z-scores!

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Contacts

• Recall
• So, for proportions, this will be
• For our bank example 0.727
• We can then use the Z table or Z area tool to
  find P(phat gt 0.3333)
• P(phatlt0.3333) 0.7664
• P(phatgt0.3333) 1-0.7664 0.2336

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Groovy!

• Groovy MMs are supposed to make up 30 of the
  candies in a bag of Groovy MMs. In a bag of 250,
  what is the probability that we get at least 25
  groovy candies?

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Misc. Notes.

• Shape of sampling distribution will become more
  normal as n increases.
• Variability (standard deviation) DECREASES as n
  increases.

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Example

• Bank believes that 7 of people who receive loans
  will not make payments on time. They recently
  approved 200 loans.
• Check assumptions/conditions
• Assume people are SRS, 200 lt10 of pop.
• np 200.07 14gt10
• nq200.93 186 gt 10
• Whats the appropriate model?
• Mean?
• SD?
• N(0.07, )
• N(0.07,0.018)

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Bank example

• What is the probability that more than 10 of the
  200 loans will not make payments on time?
• In other words, what proportion of all possible
  samples will have more than 10 not make payments
  on time?
• (draw picture)
• How can we solve this?
• Z-scores!

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Bank loans

• Recall
• So, for proportions, this will be
• For our bank example
• We can then use the Z table or Z area tool to
  find P(phat gt 0.1)
• P(phatlt0.1) 0.9525
• P(phatgt0.1) 1-0.9525 0.0475