Confidence intervals

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Confidence intervals
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Estimates and errors

• Want to estimate parameters such as m (population
  mean) or p (population proportion)
• Obtain a SRS and use our estimators, and
  Even though these are good estimators, they will
  rarely be exactly on target.
• For this reason, we typically include a margin of
  error along with the estimates, forming an
  interval.

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Margins of error

• Calculate margin of error based on info from the
  SRS and the sampling dist. of the estimator.
• The margins of error need to be larger - you will
  have to allow for the possibility of greater
  error if you
• Have a small sample
• Want higher level of confidence in your interval
• Have a large population variance

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Confidence level

• How large does the margin of error have to be so
  that you can be 100?C confident in an interval?
• Can you be 100 confident that an interval
  encloses the parameter?
• Only way is to give an interval including all
  possible values the parameter can take, such as
  the interval (0,1) for proportions - completely
  useless!
• We have to settle for less than 100 confidence.

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Confidence interval (CI) for m

• For large samples, CLT says that is approx.
  normally dist. with mean m and std. deviation
  s/sqrt(n).
• Use normal dist. and std. deviation of to
  find margins of error /- z ? s/sqrt(n).
• z is the z-score that marks off top (1-C)/2 of
  the standard normal curve
• Level C CI /- z ? s/sqrt(n)

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Example mean age of customers

• To determine the average age of its customers, a
  mens clothing manufacturer took a SRS of 50
  customers and found the average age of the sample
  was 36. If we know the standard deviation of the
  age of all the customers is 12
• Whats a 95 CI for the mean age of all
  customers?
• Suppose you want to make the 95 CI narrower, say
  /- 2 years. How large a sample is required?

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What do we mean by confidence?

• Having 95 confidence in an interval is NOT same
  as saying that there is a 95 probability that
  the parameter lies in the interval!
• Once the sample has been taken and used to
  calculate a CI, there is nothing random left- the
  interval either encloses the parameter or not.
• We ARE saying that 95 of samples will yield CIs
  that enclose the parameter