SECTION 8.1 Summary

• Inference about a population proportion is based on an SRS of size . When is large, the distribution of the sample proportion is approximately Normal with mean and standard deviation .
• The estimated standard deviation of the distribution of is the standard error of
  

• The margin of error for confidence level C is
  

  where is the value for the standard Normal density curve with area C between and .

• The large-sample level C confidence interval for is
  

  We recommend using this method when the number of successes and the number of failures are both at least 10.

• The plus four estimate of a population proportion is obtained by adding two successes and two failures to the sample and then using the procedure. We recommend using this method when the sample size is at least 10 and the confidence level is 90%, 95%, or 99%.
• Tests of are based on the statistic
  

  with -values calculated from the distribution. Use this test when the expected number of successes and the expected number of failures are both at least 10.

• The sample size required to obtain a confidence interval of approximate margin of error for a proportion is found from

  where is a guessed value for the proportion, and is the standard Normal critical value for the desired level of confidence. To ensure that the margin of error of the interval is less than or equal to no matter what may be, use

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• Power calculations for significance tests on a single proportion are easily performed using software. In addition to the characteristics of the significance test (null and alternative hypotheses, type I error), you can specify the power and determine the sample size or specify the sample size and determine the power.