From the Data at Hand to the World at Large

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Part V

• From the Data at Hand to the World at Large

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Chapter 18Sampling Distribution Models

• Modeling the Distribution of sample proportions
• From 1000 randomly selected voters (2004)
• Poll 1 John Kerry 49
• Poll 2 John Kerry 45.9

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Assumptions and Conditions

• Assumptions
• The sampled values must be independent of each
  other
• The sample size, n, must be large.
• Conditions
• 10 Condition
• If drawing without replacement then the sample n
  must be no larger than 10 of the population
• Success / Failure condition
• The sample size has to be big enough so that both
  np and nq are greater than 10

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The Sampling Distribution Model for a Proportion

• Provided that the sampled values are independent
  and the sample size is large enough, the sampling
  distribution of p is modeled by a normal model
  with mean
• And standard deviation

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Models for proportions

• Exercise 10 page 424

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Means The Fundamental Theorem of Statistics

• Central Limit Theorem (CLT)
• The sampling distribution of any mean becomes
  normal as the sample grows (independent
  observations)
• As the sample size n increases, the mean of n
  independent values has a sampling distribution
  that tends toward a normal model with mean
  equal to the population mean and standard
  deviation

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CLT
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Assumptions and Conditions

• Random Sampling Condition
• The values must be sampled randomly
• Independence assumption
• 10 condition
• The sample size is less than 10 of the population

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Exercise

• Step-by-step page 418
• Ex.36 page 426

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Standard Error

• When we estimate the standard deviation of a
  sampling distribution using statistics found from
  the data, the estimates are called standard
  error
• For a proportion
• For the sample mean

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Dont confuse the sampling distribution with the
distribution of the sample

• Distribution of the sample
• Take a sample
• Look at the distribution on a histogram
• Calculate summary statistics
• Sampling Distribution
• Models an imaginary collection of the values that
  a statistic, might have taken from all the
  samples that you didnt get.
• We use the sampling distribution model to make
  statements about how statistics varies

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Confidence Intervals for Proportions

• Example Infected Sea fan corals at Las Redes
  Reef (LRR)

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Confidence Intervals

• 68 of the samples will have p within 1 SE of p.
  And 95 of all samples will be within p2SE
• We know that for 95 of random samples p will be
  no more than 2SE away from p.
• Now from p point of view, there is a 95 chance
  that p is no more than 2SE away from p

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Confidence interval

• We are 95 confident that between 42.1 and 61.7
  of LRR sea fans are infected.
• Margin of Error
• Certainty vs. Precision
• Estimate M.E.
• The margin of error for our 95 confidence
  interval was 2SE
• For 99.7 confident 3SE
• 100 Confident 0 to 100
• Low Confidence 51.8 to 52.0

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Critical Values z

• The number of standard errors to move away from
  the mean of the sampling distribution to
  correspond to the specified level of confidence.
• Find z (critical value) for 98 confidence.
• For 95?

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Confidence interval(one-proportion z-interval)

• The critical value z depends on the particular
  confidence interval we specify and
• Assumptions
• Independence
• Conditions
• Randomization
• 10 Condition

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Exercise

• 13 Page 444

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Chapter 20 Testing Hypothesis about proportions

• Example
• Metal Manufacturer
• Ingots
• 20 defective (cracks)
• After Changes in the casting process
• 400 ingots and only 17 defective
• IS this a result of natural sampling variability
  or there is a reduction in the cracking rate?

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Hypotheses

• We begin by assuming that a hypothesis is true
  (as a jury trial).
• Data consistent with the hypothesis
• Retain Hypothesis
• Data inconsistent with the hypothesis
• We ask whether they are unlikely beyond
  reasonable doubt.
• If the results seem consistent with what we would
  expect from natural sampling variability we will
  retain the hypothesis. But if the probability of
  seeing results like our data is really low, we
  reject the hypothesis.

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Testing Hypotheses

• Null Hypothesis H0
• Specifies a population model parameter of
  interest and proposes a value for this parameter
• Usually
• No change from traditional value
• No effect
• No difference
• In our example H0p0.20
• How likely is it to get 0.17 from sample
  variation?