Unit 13

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Unit 13

• Confidence Intervals for Proportions

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Smoking

• There is concern that the proportion of college
  students who smoke cigarettes is increasing. In a
  random sample of 100 college students 23
  identified themselves as smokers
• What does this say about the population
  proportion, p, of college age smokers?
• We have a sample proportion,

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Sampling Distribution of

• Recall what we know about how varies from
  sample to sample
• If n is large has an approximately normal
  distribution
• It has a mean of p
• It has a standard deviation of

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The Standard Error

• Unfortunately if we do not know p we cannot
  evaluate
• We must instead evaluate an expression called the
  standard error of sample proportion

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Great!

• We now know that about 95 of all samples of 100
  will have within
• p 2SE and p 2SE
• p - .0824 and p .0824
• Stated alternatively the interval
• .23 - .0824 to .23 .0824 .1476 to .3124
• has a 95 probability of capturing the true
  proportion, p, of college students who smoke

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Congratulations!

• You have just calculated you first confidence
  interval
• We announced that we are 95 confident that
  between 14.76 and 31.24 of all college students
  smoke
• Our 95 confidence interval takes the form

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Margin of Error

• The expression
• is called the margin of error, ME
• Confidence intervals for a proportion look like
• estimate /- ME

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More Confident?

• If we wanted to be more confident, say 99.7
  confident, the margin of error would have to be 3
  SE
• The more confident we want to be, the larger our
  margin of error
• To be 100 confident all we can announce is that
  the proportion of college students who smoke is
  between 0 and 100 - not very informative!

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Tension

• There is a tension between certainty and
  precision
• The most common confidence levels are 90, 95
  and 99.

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Critical Values

• To evaluate a 95 confidence interval we used 2
  SE to find the ME
• To evaluate a different confidence level we need
  to change the number of SEs.
• This number of SEs is called a critical value

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The Precise Critical Value

• We have been using 2 for the critical value for a
  95 confidence interval
• If we use a computing device we realize the
  precise z score for trapping the middle 95 of
  the standard normal curve is 1.96

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How About 99 Confidence?

• What is the critical value, z-score, for 99
  confidence? This would be the 99.5 percentile
• Critical value 2.576

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And 90 Confidence?

• What is the critical value, z-score, for a 90
  confidence interval? This would be the 95th
  percentile.
• Critical value 1.645

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One Proportion Confidence Interval

• Provided the sample size is large the confidence
  interval for a population proportion can be
  calculated using
• Where z is the appropriate critical value

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Catalog Sales

• A catalog sales company promises delivery of all
  sales made on its internet site within 3 business
  days. A random sample of 200 sales showed that
  176 were delivered within 3 business days.
• Establish and interpret both 95 and 99
  confidence intervals for the proportion of sales
  delivered within 3 business days

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Solution

• We are 95 confident that between 83.5 and 92.5
  of orders are received within 3 business days
• We are 99 confident that between 82.1 and 93.9
  of orders are received with 3 business days

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Margin of Errors

• Note that the margin of error increased from 4.5
  to 5.9 in the previous example as the confidence
  interval changed from 95 to 99
• If your margin of error is too large to be
  practically useful you have 2 options
• Reduce you confidence level
• Increase your sample size

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Legal Music

• A random sample of 168 students were asked how
  many songs in their digital library were legally
  purchased. Overall the students reported a total
  of 117079 songs of which 23.1 were legal.
• Construct and interpret a 98 confidence interval
  for the proportion of legal downloads of digital
  music.
• Have any conditions of a binomial experiment been
  violated? Explain.

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Solution

• We can be 95 confident that between 22.8 and
  23.4 of music downloads by students are legal
• The incredibly small ME is due to the large
  sample size
• The trials in this experiment are not
  independent. A student who illegally downloads
  one song will likely download another illegally