Estimating proportions with confidence

1
Estimating proportions with confidence

• Utts Ch 20

2
Confidence Interval

• Def an interval of values computed from sample
  data that is almost sure to cover the true
  population number

3
A Confidence Interval for a Proportion
To be exact, we would use 1.96(SD) instead of
2(SD). However, rounding 1.96 off to 2.0 will not
make much difference.
Problem The true proportion SD depends on p,
which is estimated with p-hat. Solution
Substitute the sample proportion for the standard
deviation-- called the standard error (SE).
4
Levels of Confidence

• 95 confidence interval sample proportion
  1.96(SD)
• 90 confidence interval sample proportion
  1.645(SD)
• 99 confidence interval sample proportion
  2.576(SD)

In 95 of all samples, the true proportion will
fall within plus/minus 2 standard deviations of
the sample proportion.
5
How many people/things do I need in my sample?
(i.e. How much money do I need for my project)
6
Derivation of the Margin of Error
Two formulas for a 95 confidence interval
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Confidence Intervals from the Media

• Formula for a 95 confidence interval
• sample proportion margin of error

8
Poll Youth Find Many Uses for Cell PhonesBy
WILL LESTER, Associated Press Apr 3, 2006Young
adults and minorities are leading a revolution in
how Americans use their cell phones. People from
age 18 to 29 and minorities are more likely to
use their phones as personal computers, digital
music players, cameras and more, an AP-AOL-Pew
poll found.

• 28 percent, said they sometimes don't drive as
  safely as they should because they are using a
  cell phone.
• 36 percent, said they are sometimes shocked at
  the size of their service bill.
• Almost nine in 10 users of cell phones say they
  encounter others using those phones in an
  annoying way.
• Only 8 percent of cell users acknowledge their
  own use of cell phones is sometimes rude.

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Margin of error
Interviewing for the survey was conducted by
telephone March 8-28, 2006 among a sample of
1,503 adults age 18 and older. Approximately half
of the interviews (752) were conducted using a
landline number frame, with the remainder
conducted from a cell phone number frame (751).
For results based on the total sample, one can
say with 95 confidence that the error
attributable to sampling is plus or minus 3
percentage points. Sampling error for subgroups
in the sample would be larger than 3 points. In
addition to sampling error, one should bear in
mind that question wording and practical
difficulties in conducting surveys can introduce
error or bias into the findings of opinion polls.
10
A public opinion poll revealed that 54 of
respondents agreed that gay and lesbian couples
could be good parents. Source Pew Research
Center for the People and the Press survey of
1,515 U.S. adults, Oct 15-19, 2003 margin of
error 3

• A 95 CI for the population proportion is 51 -
  57. This means that
• 95 of the population agrees that gay and lesbian
  couples could be good parents between 51 and 57
  of the time.
• There is a 95 probability that a randomly
  selected person agrees that gay and lesbian
  couples could be good parents
• We are 95 confident that between between 51 and
  57 of the population agrees that gay and lesbian
  couples could be good parents .
• There is a 5 probability between between 51 and
  57 of the population agrees that gay and lesbian
  couples could be good parents .