Estimating a Population Proportion

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Estimating a Population Proportion

• Goal Given a sample proportion, estimate the
  value of the population proportion p.
• Example In a sample of 750 people, 27 said
  they feel that health care is the most important
  issue facing our state. What proportion of the
  population feels that health care is the most
  important issue?

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Assumptions

• The sample is a simple random sample.
• The conditions for the binomial distribution
  apply There are a fixed number of trials, the
  trials are independent, there are two categories
  of outcomes, and the probabilities remain
  constant for each trial.
• The normal distribution can be used to
  approximate the distribution of sample
  proportions, since and
•  
• Since p and q are not known, we use the sample
  proportion to estimate their values.

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New Notation

• p population proportion
• sample proportion (of x successes in a
  sample of size n)

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The sample proportion is the best point
estimate (single value approximation) of the
population proportion p.   Problem Using
to approximate p doesnt convey how accurate we
expect our estimate to be. To do that, we need
confidence intervals
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Confidence Intervals (CI)

• A confidence interval is a range (or interval) of
  values used to estimate the true value of the
  population parameter.
•  
• A confidence level is the probability that our
  confidence interval contains the true value of p.
•  
• The confidence level is expressed as a
  probability 1- a

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Common Values

• 90 confidence level (a 0.10)
• 95 confidence level (a 0.05)
• 99 confidence level (a 0.01)

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An example of a Confidence Interval

• Based on our survey earlier,
• The 95 confidence interval estimate of the
  population proportion p is 0.235 lt p lt 0.305
•  
• This means that there is 95 chance that this
  interval contains the actual population
  proportion p.
•  
• In other words, 95 of the time that we do a
  sample, the confidence interval will contain the
  true population proportion.

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Critical Values

• A critical value is a z-score that separates
  outcomes that are likely to occur from those that
  are unlikely to occur.
•  
• An example For 95 confidence interval

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For the 95 confidence interval, a .05 Notice
that 0.025 falls above the critical value, and
0.025 falls below the opposite (negative)
critical value. Each of these areas is
a/2.   Notation The critical value za/2 is the
positive z-value that separates the top area of
a/2. -za/2 is the boundary of the bottom area of
a/2.
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Another Example

• So if our confidence level was 99, the critical
  value za/2 would be the score that separates the
  top 0.5 of data, and za/2 would separate the
  bottom 0.5 of data.
• Leaving 99 of the data between za/2 and za/2

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Finding Critical Values

• Example
• For the 95 confidence interval, the area above
  za/2 is .025, so the area below is 1-.025 .975
• So P(z lt za/2) .975. From the table, we find
  za/2 1.96

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Common Critical Values

• 90 a .10 Critical value 1.645
• 95 a .05 Critical value 1.96
• 99 a .01 Critical value 2.575
•  
• (listed at bottom of z-score table)

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Creating a Confidence Interval
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Margin of Error

• The margin of error E is the maximum likely (with
  probability 1-a) difference between the observed
  proportion and the population proportion p.

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Summary of Procedure for finding a Confidence
Interval

• Verify the required assumptions are satisfied
• Find the critical value that corresponds to the
  desired confidence level
• Evaluate the margin of error E
• Find the values and .
  Write the confidence interval
• Round values to three decimal places

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Example

• In a sample of 750 people, 27 said they feel
  that health care is the most important issue
  facing our state.
•  
• 95 confidence level, so

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Example continued

• so our confidence interval is
• 0.238 lt p lt 0.302

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Example continued

• 0.238 lt p lt 0.302
• Based on our survey results, we are 95 confident
  that the true percentage of Washingtonians who
  feel that health care is the most important issue
  facing the state is between 23.8 and 30.2.

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From a Confidence Interval

• If you know a confidence interval, the middle
  value is the point estimate (in this case, ).
  You can find it by calculating
• If you know a confidence interval, the Margin of
  Error is half the width of the confidence
  interval. You can find it by calculating

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Determining Sample Size from desired Margin of
Error
when is known
when is not known
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Homework

• 6.2 5, 13, 17, 21, 27, (29)