12'2 Inference for a Population Proportion

1
12.2 Inference for a Population Proportion

• We are interested in the unknown proportion p of
  a population that has some outcome call the
  outcome we are looking for a success.
• The statistic that estimate the parameter p is
  the sample proportion
• p-hat count of successes in sample
• count of observations sample

2
Recall The sampling distribution of p-hat

• The mean of the sampling distribution is p.
• The sample proportion p-hat is an unbiased
  estimator of the population proportion p.
• The standard deviation of p-hat is

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Solution to not knowing p

• ?To test the null hypothesis that the unknown
  p has a specific value , replace p by
  in the test statistic
• to get
• ?In a confidence interval for p will be
  close to p if n is large, so replace the std.
  dev. by the standard error of .

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Assumptions for inference about a proportion

• The data are an SRS from the population of
  interest
• The population is at least 10 times as large as
  the sample.
• For a test of , the sample size n
  is
• so large that and
• For a confidence interval, n is so large that
• and
• (successes) (failures)

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Example

• A coin that is balanced should come up heads half
  the time in the long run. The population for coin
  tossing contains the results of tossing the coin
  forever. The parameter p is the probability of a
  head, which is the proportion of all tosses that
  give a head. The tosses we actually make are an
  SRS from this population. The French naturalist
  Count Buffon (1707-1788) tossed a coin 4040
  times. He got 2048 heads. The sample proportion
  of heads is 0.5069. That is more than one-half.
• Is this evidence that Buffons coin was not
  balanced?
• Find the 95 confidence interval.

6
Recall n for a desired margin of error

• To determine the sample size n that will yield a
  level C confidence interval for a population
  proportion p, solve the following for n
• is the standard normal critical value for
  level of confidence we want
• is a guess (or is from a study).

7

• Many colleges that once enrolled only male or
  only female students have become coeducational.
  Some administrators and alumni were concerned
  that the academic standards of the institutions
  would decrease with the change. One formerly
  all-male college undertook a study of the first
  class to contain women. The class consisted of
  851 students, 214 of whom were women. An
  examination of first-semester grades revealed
  that 15 of the top 30 students were female.
• What is the proportion of women in the class?
  Call this value p-nought.
• Assume that the number of females in the top 30
  is approximately a binomial random variable with
  n30 and unknown probability p of success. In
  this case success corresponds to the student
  being female. What is the value of p-hat?
• Are women more likely to be top students than
  their proportion in the class would suggest?
  State hypotheses that ask this question, carry
  out a significance test, and report your
  conclusion in non- technical language.

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Methods of Poll Explained

• The Times-Dispatch/12 News poll was conducted by
  the research department of Medial General, Inc.,
  parent company of the Times-Dispatch. Based on
  telephone interviews October 23 through Wednesday
  with 502 respondents who identified themselves as
  registered voters, the survey had a sampling
  error of plus or minus 4.5 percentage points. In
  other words, one could say with 95 percent
  certainty that the results of the poll would vary
  4.5 percent in either direction if the entire
  adult population of Virginia had been polled.
• Verify that the newspapers margin of error is
  correct. What is the exact margin of error?