Chapter 19: Confidence Intervals for Proportions

1
Chapter 19Confidence Intervals for Proportions

• Far better an approximate answer to the right
  question,than an exact answer to the wrong
  question.
• -John W. Tukey

2
Standard Error

• To find the standard error
• Because the sampling distribution model is
  Normal
• 68 of all samples will be within
• 95 of all samples will be within
• 99.5 of all samples will be within

3
Confidence Interval

• One-proportion z-interval
• Putting a number on the probability that this
  interval covers the true proportion.
• Our best guess of where the parameter is and how
  certain we are that its within some range.

4
Margin of Error

• The extent of the interval on either side of
• is called the margin of error (ME).
• In general, confidence intervals are written as
• There is a conflict between certainty and
  precision
• Choose a confidence level the data does not
  determine the confidence level

5
Assumptions and Conditions

• Independence Assumption
• The data values are assumed to be independent
  from each other.
• Plausible independence condition
• Do the data values somehow affect each other?
• Dependent on knowledge of the situation
• Randomization condition
• Where data sampled at random or generated from a
  properly randomized experiment?
• Proper randomization helps ensure independence
• 10 condition
• Samples are always drawn without replacement
• Samples size should be less than 10 of the
  population

6
Assumptions and Conditions

• Sample Size Assumption
• -Based upon the Central Limit Theory (CLT)
• The sample must be large enough to make the
  sampling model for the sampling proportions
  approximately Normal.
• More data is needed as the proportion gets closer
  to either extreme, 0 or 1.
• Success/failure condition expect at least 10
  successes and 10 failures.

7
One-proportion z-interval

• When the conditions are met, we are ready to find
  the confidence interval for the population
  proportion, p. Since the standard error of the
  proportion is estimated by

8
TI-83 Tips

• TI-83 can calculate a confidence interval for a
  population proportion.
• STAT
• TESTS
• A 1-PROPZInt

9
TI-83 Tips

• Enter the number of successes observed and the
  sample size.
• Specify a confidence level and then Calculate.

10
Caution! Caution! Caution!

• Dont mistake what the interval means
• Do not suggest that the parameter varies.
• The population parameter is fixed the interval
  varies from sample to sample.
• Do not claim that other samples will agree with
  this sample.
• The interval isnt about sample proportions it
  is about the population proportion.
• Dont be certain about the parameter.
• We cant be absolutely certain that the
  population proportion isnt outside the interval
  just pretty sure.

11
Caution! Caution! Caution!

• Dont forget its a parameter.
• The confidence interval is about the unknown
  population parameter, p.
• Dont claim too much.
• Write your confidence statement about your
  sample.
• Take responsibility.
• Confidence intervals are about uncertainty. You
  are uncertain, however, not the parameter.

12
Margin of Error Too Large to be Useful?

• Think about the margin of error during design of
  the study.
• Choose a larger sample to reduce variability in
  the sample proportion.
• To cut the standard error (and the ME) in half,
  quadruple the sample size.
• Remember, though, that bigger samples cost more
  money and effort.

13
Margin of Error An Example

• Suppose a candidate is planning a poll and wants
  to estimate voter support within 3 with 95
  confidence. How large a sample is needed?

14
Violation of Assumptions

• Watch out for biased samples.
• Check potential sources of bias.
• Relying on voluntary response
• Undercoverage of the population
• Nonresponse bias
• Response bias
• Think about independence.