Confidence Intervals for Proportions

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Chapter 19

• Confidence Intervals for Proportions

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Sampling Distribution Models
Population Parameter? p
Population
Inference
Sample
Sample Statistic
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Sampling Distribution of

• If our two conditions hold then we know
• Shape Approximately Normal
• Center The mean is p.
• Spread The standard deviation is

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Sampling Distribution of

• Recall the two conditions
• 10 Condition The size of the sample should be
  less than 10 of the size of the population.
• Success/Failure Condition np and n(1 p) should
  both be greater than 10.

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Sampling Distribution of

• So If we know the population proportion p and the
  sample size is big enough then we can
  intelligently think about possible by
  using the normal model.
• We can find the probability of obtaining a
  particular
• We can determine if observing a particular is
  unlikely or not.

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Sampling Distribution of

• For example by using the 68-95-99.7 Rule we can
  say something like this
• 95 of the time the sample proportion, will
  be between

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Inference

• Unfortunately the population parameter, p, is
  usually unknown. We would like to use a sample
  to tell us something about p.
• Use the sample proportion, , (as our best
  guess) to make inferences about the population
  proportion p.

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War on Terrorism

• According to a March 2nd 2006 ABC News/Washington
  Post poll of 1,000 adult Americans, 46 of those
  surveyed disapprove of the way that Bush is
  handling the US campaign against terrorism.
• This poll was conducted by The Washington Post,
  so lets assume (hope) that they randomized
  correctly and obtained a representative sample.
• What is the population here?
• What is the population parameter?

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War on Terrorism

• So what is the sampling distribution of the
  proportion of US adults who disapprove of the way
  that Bush is handling the US campaign against
  terrorism.?
• We know?
• n 1,000
• .46
• if conditions
  hold.(do they?)

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War on Terrorism

• What dont we know?
• p - we dont know the actual proportion of US
  adults who disapprove of the way that Bush is
  handling the US campaign against terrorism.
• Since we dont know p then we also dont know
• What can we do?
• We can use to find an estimate of

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Estimation

• We expect that p and should be similar so we
  can use to estimate
• When we use to estimate the standard
  deviation, this is called the standard error of

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What does this tell us?
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What does this tell us?

• Once again using the 68-95-99.7 rule we know
  that
• About 68 of the time (i.e. for about 68 of
  random samples), will be no more than 1
  away from p.
• About 95 of the time (i.e. for about 95 of
  random samples), will be no more than 2
  away from p.
• About 99.7 of the time (i.e. for about 99.7 of
  random samples), will be no more than 3
  away from p.

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What does this tell us?

• Lets think about the second interval (95).
• Start with (because we know this value) and go
  out about 2 in either direction.
• We can be 95 sure (confident) that the interval
  will contain p.

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War on Terrorism

• About 95 of the time (i.e. for about 95 of
  random samples), will be no more than 0.032
  away from p
• Start at 0.46 and go out about 0.032 in
  each direction
• We can be 95 confident that this interval will
  contain p.
• We are 95 confident that the true proportion of
  adults who are dissatisfied with the way the war
  on terrorism is going is between 32 and 38.

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Interpretation

• Plausible values for the population parameter p.
• 95 confidence in the process that produced this
  interval.

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Statistical Confidence

• Two things can happen when we create the interval
  as above
• p can either be in the interval (which will
  happen in about 95 of the intervals).
• p can be outside the interval (which will happen
  in about 5 of the intervals).
• One thing that cant happen
• The parameter value cant change!!

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Statistical Confidence

• We dont know which is true.
• So, we rely on our statistical confidence.
• The best we can say is, We are 95 confident
  that the true population proportion lies within
  the interval we construct.

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Statistical Confidence

• WE ARE NOT SAYING that p is in our interval 95
  of the time.
• The above phrase implies that p is moving
  around which we already said this cannot happen
  (remember p is some unknown fixed value).
• If we were to calculate lots of intervals, the
  population parameter will be in about 95 of
  them.

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Confidence Intervals

• Confidence intervals come from the fact that we
  could take multiple samples and calculate
  multiple 95 confidence intervals and, if we were
  using the same method to find all the intervals,
  we would expect that about 95 of the intervals
  we constructed would contain the true parameter
  (population proportion).

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95 Confidence

• If one were to repeatedly sample at random 1000
  registered voters and compute a 95 confidence
  interval for each sample, 95 of the intervals
  produced would contain the population proportion
  p.

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Confidence Intervals

• So, what does the interval look like?
• Confidence intervals for the population
  proportion have the form
• For 95 confidence intervals,

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Confidence Intervals

• ME is the margin of error
• The extent of the interval on either side of our
  estimate
• In general,
• where is called a critical value.

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Construction of CI

• , the point estimate, is the center of the
  interval. It merely shifts the interval along
  the axis.
• , the critical value, is the number of
  multiples of the standard error needed to form
  the desired CI. This will depend on the level of
  confidence you want.

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How to find

• Need z-tables
• Based on Normal model
• Between what two z-values do 95 of the
  observations lie on N(0,1)?

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How to find

• The z-values for a 95 confidence interval are
  not exactly 2 and 2.
• We use these numbers as an approximation.
• 1.96 and 1.96 are more exact.
• So, a 95 CI for the population proportion looks
  like

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How to find

• What does a 99 CI for p look like?
• What does a 90 CI for p look like?
• What does an 80 CI for p look like?

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Construction of CI

• , the standard error, is the estimate for
• , the margin of error, is ½ the width
  
• of the CI. This merely determines the width of
  the interval.
• What happens to ME if n increases?

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Construction of CI

• So, the CI for p looks like

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Confidence Intervals

• Now we know what the interval looks like, but how
  do we know we can do all this?
• It is based on the Normal model
• Did we check any assumptions beforehand?
• What were the assumptions needed for sampling
  distribution for sample proportion?

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Confidence Intervals

• We dont know p or q.
• So, check the following assumptions
• Random sample
• Were data sampled randomly or are they from a
  randomized experiment?
• Independence
• Do data values affect one another?
• n lt 10 of population size
• Success/Failure

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Step to Forming a CI

• 1) Describe the population parameter of
  concern.
• Ex p proportion of adults dissatisfied with
  the way the war on terrorism is going
• 2) Specify the confidence interval criteria
• a) check assumptions
• random sample
• independence
• n lt 10
• success/failure

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Steps to a Confidence Interval

• b) state the level of confidence
• c) determine the critical value, z
• 3) Collect and present sample information
• a) collect the data from the population
• b) find the point estimate,
• 4) Determine the confidence interval
• a) find the standard error,

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Steps to a Confidence Interval

• b) find the margin of error,
• c) find the interval,
• d) describe your results interpret the interval
• I am ___ confident that the true population
  proportion falls within the interval I
  constructed.

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Example

• Ch. 19 7
• True or False?
• For a given sample size, higher confidence means
  a smaller margin of error.
• For a specified confidence level, larger samples
  provide smaller margins of error.
• For a fixed margin of error, larger samples
  provide greater confidence.
• For a given confidence level, halving the margin
  of error requires a sample twice as large.

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Example

• For a given sample size, higher confidence means
  a smaller margin of error.
• Solution
• ME z(SE( )) z
• -fixed n implies fixed SE
• -higher confidence implies higher z
• (see Table T)
• -so, with fixed SE and increasing z, ME
  increases, the statement is FALSE

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Example

• b) For a specified confidence level, larger
  samples provide smaller margins of error.
• Solution
• ME z(SE( )) z
• -certain confidence interval fixed z
• -bigger sample increasing n implies smaller
• -so, with fixed z and decreasing , ME
  decreases, the statement is TRUE

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Example

• c) For a fixed margin of error, larger samples
  provide greater confidence.
• Solution
• ME z( ) z
• -fixed ME
• -larger samples imply smaller
• -so for ME to remain the same, z must increase
• -increasing z implies larger confidence, so the
  statement is TRUE

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Example

• d) For a given confidence level, halving the
  margin of error requires a sample twice as large.
• Solution
• ME z( ) z
• -given confidence level implies fixed z
• -halving the margin of error means dividing ME
  by 2
• -if you divide one side by 2, must divide the
  other by 2
• -so, if you divide ME by 2, you need to multiply
  the sample size by 4, not 2, the statement is
  FALSE

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Example

• Ch. 19, 20
• A city ballot includes a local initiative that
  would legalize gambling. The issue is hotly
  contested and two groups decide to conduct polls
  to predict the outcome. The local newspaper
  finds that 53 of 1200 randomly selected voters
  plan to vote yes, while a college statistics
  class finds 54 of 450 randomly selected voters
  are in support. Both groups will create 95
  confidence intervals.

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Example

• a) Without finding the confidence intervals,
  explain which one will have the larger margin of
  error.
• Because the classes sample size is smaller, its
  interval will be larger.

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Example

• b) Find both confidence intervals.
• Newspaper (50.2, 55.8)
• We are 95 confident that the true proportion of
  people who will vote to legalize gambling is
  between 50.2 and 55.8.
• Class (49.4, 58.6)
• We are 95 confident that the true proportion of
  people who will vote to legalize gambling is
  between 49.4 and 58.6.

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Example

• c) Which group concludes that the outcome is too
  close to call? Why?
• The students should conclude that their interval
  is too close to call because 50 is in the
  interval, meaning that it is quite likely that p
  could be 50.

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Cautions about Confidence Intervals

• Do NOT suggest that the parameter p varies!
• Do NOT imply you are certain about the parameter
  p!
• Be sure to remember that the confidence interval
  is about the parameter, NOT the sample
  proportion(s)!

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Sample Size and the ME

• How precise should our margin of error be?
• We know that we cannot be exact, but we dont
  want our margin of error to be too large.
• If it is too large, it may not be useful.

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Sample Size and the ME

• There are two ways to adjust our ME.
• You can reduce your confidence level.
• As you reduce confidence, the value of z
  decreases.
• However, confidence levels less than 80 are
  rarely used in real studies. 95 and 99 are more
  common.
• You can change your sample size.
• If we look at the equation for ME, we see that
  changing the sample size will change ME.
• In many cases, we may want to know how large of a
  sample we should take to guarantee a certain ME.

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Sample Size and the ME

• Determining sample size.
• We know that ME z
• We can manipulate this equation with algebra.

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Sample Size and the ME

• This will allow us to calculate the minimum
  sample size needed to have a certain margin of
  error.
• The worst case scenario, the one that needs the
  largest sample size, is when p 0.5. So, if we
  use this value for , we will be safe, meaning
  that we wont choose a sample size too small to
  meet our required margin or error.
• If you get a decimal, always round up!

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Sample Size and the ME

• Example
• Suppose that we want to estimate the proportion
  of ISU students who like Stat 101 within 3 with
  95 confidence. How large of a sample size is
  needed?
• ME 0.03, z 1.960, and 0.5