Inference for Proportions

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Section 8.1

• Inference for Proportions

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Preface (text notes)
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Preface (text notes)
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Categorical Data

• Up to this point we have only talked about
  quantitative variables.
• Many studies are about variables such as race,
  sex, occupation, make of a car, smoking or
  non-smoking, type of complaint received, etc.
• Under these types of variables our data are
  counts or percents obtained from counts
• How do we numerically summarize numerical data?

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Big Picture
Population Parameter? p
Population
Inference
Sample
Sample Statistic
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Cold outside?

• Do you think the temperature is low outside?
• There are two possible answers to this question
  Yes or No (at least by our definition).
• Pretend that out of a sample of 20 people in our
  class, 15 say yes.
• So the proportion of people who walk outside and
  curse, Its ((_at_( cold out here is
• 15 0.75 or 75
• 20

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Cold!

• Lets assume that Stat 226 section D is a good
  representation of the ISU population
• We are interested in knowing the opinion of the
  entire ISU student body
• This is a new parameter that is denoted by P
• Thus P represents the population proportion
• Recall that we use a statistic to estimate the
  population parameter of interest
• In this case 0.75 is a statistic or simply
  is the sample statistic.

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Sample Proportion

• In general
• A sample proportion from an SRS of size n is the
  number of successes over the total sample size

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Sampling Distribution of

• Recall that a sampling distribution of a
  statistic is the distribution or shape center and
  spread of that are obtained from all
  possible samples of the same size.

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Sampling Distribution of

• As the sample size increases, the sampling
  distribution of becomes approximately normal
  with
• Mean p (Thus is an unbiased estimator)
• Standard Deviation
• Where p is the true population proportion
• This is denoted as

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Standard Error

• Since we dont know what p is we can use in
  the standard deviation to get the standard error.
• So we have

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One Issue

• Unfortunately in practice it has been shown that
  using the estimate to construct
  confidence intervals can be inaccurate
• For example lets suppose that not one person
  noticed that the temperature outside is low (all
  wearing shorts?).
• Then and which means that we
  are certain that no one thinks the temperature is
  low
• We need to find an estimator that is still
  unbiased but doesnt have this characteristic

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Wilson Estimate

• Wilson estimate of p (the population proportion)
  is
• We are essentially adding four observations and
  two of them are considered successes (and two
  failures). The Wilson estimate always keeps us
  away from 0 and 1 and works well in practice.
• So the Wilson estimate for the proportion of
  people for whom the temperature outside is low

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Confidence Interval for p

• So a Confidence Interval for p based on the
  Wilson estimate is
• Where the standard error is
• And z is the value of the standard Normal
  density curve with area C between -z and z
• The margin of error is
• Same old method in a different hat (or tilda)
• estimate ? margin of error
• Caution Use this interval when sample size is
  at least n5 and for confidence level 90, 95,
  or 99

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Cold! (again)

• A 95 confidence interval for the proportion of
  ISU students who think the temperature outside is
  low
• So the 95 CI is (0.526, 0.889)
• Interpretation
• I am 95 confident that between 52.6 and 88.9
  ISU students first thought when they head
  outside is the temperature outside is low

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Example cont.

• What does 95 confident tell us?
• This interval is fairly wide. Hence, the
  interval does not give enough information about
  what p is. How can we make the interval narrower?

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Determine a Sample Size

• Can we find the sample size needed to get a
  confidence interval for p that has a
  pre-determined a level of confidence and margin
  of error? Yes, we can!
• Just solve for n in the margin of error equation

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Determine a Sample Size

• Recall
• Solving for n we get
• But there is an issue!
• We dont know until after data is collected.

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Determine a Sample Size

• There are two solutions to the problem
• Use a guesstimate obtained from previous studies
• Use this estimate is conservative and
  will give the desired ME and confidence level
  always, but if then sample size will
  be larger than needed and more money will be
  spent than necessary.
• So if you are not given a guesstimate, revert to

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Example

• Is there interest in a new product? One of your
  employees has suggested that your company develop
  a new product. You decide to take a random
  sample of your customers and ask whether or not
  there is interest in the new product. The
  response is on a 1 to 5 scale, with 1 indicating
  definitely would not purchase 2, probably
  would not purchase 3, not sure 4, probably
  would purchase 5, definitely would purchase.
  For an initial analysis, you will record the
  responses 1,2,and 3 as no and 4 and 5 as Yes.
  What sample size would you use if you wanted a
  95 margin of error to be 0.1 or less?

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Solution

• We want ME (or m) 0.1 and our level of
  confidence to be 95 so z1.96. And since three
  responses are no we will use
• So
• And n89 (e.g., 92.19 rounded up to 93 then
  subtract 4 89)

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Summary
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Summary
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Summary