Confidence Intervals for Proportions

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Confidence Intervals for Proportions
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Surveys and experiments often produce counts
which we can turn into proportions. Count
frequency f and compute proportion p f / N
Example p 100/600 .60 You can multiply by
100 to get a percentage .60100 60
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Sampling Error for Proportions

• Proportion in a sample is not the same as the
  true population proportion.
• We can estimate Confidence Intervals for
  proportions just like means

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The Formulas

• CI for Proportions
• CI p Z (SE)

• CI for Means
• CI X Z (SE)

The Main Difference
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p proportion 1- p not p If proportion of
people favoring abortion rights is p, then the
number of people opposing abortion rights is 1p.
If the sample size (N) is large enough the
sampling distribution will be normal. The
sampling distribution from which you are drawing
your one sample will approximate a normal
probability distribution, that is a Z
distribution when Np gt 10 and N(1 p) gt
10 If Np lt 10 or N(1-p) lt 10, then we must
use something else.
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Steps in locating the unknown population
proportion within a margin of error around our
sample score
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Example Would a majority of California voters
favor the recall election? Can you be 95
confident that more than half the voters would
approve recall. Results of the survey conducted
before the election were 89 of 150 Californians
favored recalling Gray Davis. Step 1 Generate
Proportion p f / N 89 / 150 0.593 or
59.3
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Step 2 Level of significance. Set up problem
for 95 confidence interval Step 3 Find the
Critical Value of Z (Z table or
bottom line of t table). Step 4 Plug in the
numbers
with
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A National SRS poll of N500 finds that 330 in
the sample favor stronger gun controls. Stated
in percent, 66 of this sample favors stronger
gun controls.
Step 1 What is the problem? What is the
percent of people in the population favoring gun
control. Convert the frequency into a
proportion 330/500.66
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Step 2 Set the confidence level. We choose to
take a 5 risk therefore we construct a 95
Confidence Interval. Step 3 Find the critical
Value of Z Step 4 Plug in the Numbers 95
CI .66 1.96(.02) .66 .0392 Step 6
Interpret. The mean support for gun control is
66 3.92
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As the SRS becomes bigger the estimated error
around the measure of central tendency gets
smaller. The larger the sample the less the
chance of getting an atypical average.
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Choosing between t and Z for Confidence Intervals

• Proportions Always use Z (though again, given a
  large enough N and p not too close to 0 or 1)
• Means
• Use Z if s is known, which rarely happens.
• Thus, for means use use t.
• Why? If the sample size is large enough to use
• Z, then the t table will give you the right value
  
• anyway.

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Summary of Estimation Methods for Means and
Proportions
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• Another example A national television network
  samples 1400 voters after each has cast a vote in
  a state gubernatorial election. Of these 1400
  voters, 742 claim to have voted for the
  Democratic candidate and 658 for the Republican
  candidate. There are only two candidates in the
  election.
• Assuming that each sampled voter actually voted
  as claimed and that the sample is a random sample
  from the population of all voters, is there
  enough evidence to predict the winner of the
  election? Base your decision on a 95 confidence
  interval.
• Base your decision on a 99 confidence interval.
  Explain why it requires greater evidence to make
  a prediction when we require greater confidence
  of being correct.

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First step

• Determine point estimates by converting
  frequencies into proportions and finding the
  standard error.
• Proportion of voting Democrat
• 742/1400 .53
• Standard Error

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Next Steps

• Set up the problem for a 95 CI
• Find the z score (for a .05, z1.96)
• Plug in the numbers
• .53 1.96 (.0133) (.504, .556)
• Since our interval is over .50, we can be
  reasonably confident that the Democratic
  candidate is the winner (though just barely).

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What about at the 99 interval?

• Are we that certain?
• For a 99 Confidence interval, z2.58
• .53 2.58(.0133) ? (.496, .564)
• Since our interval includes .5 (and goes slightly
  below), we cannot confidently determine a winner
  based on this sample.