Calculations based on Variability and standard error of the Sample Proportion

Variability and standard error of the Sample Proportion

To construct a confidence interval for a sample proportion, we need to know the variability of the sample proportion. This means we need to know how to compute the standard deviation or the standard error of thesampling distribution.

• Suppose k possible samples of size n can be selected from the population. The standard deviation of the sampling distribution is the "average" deviation between the k sample proportions and the true population proportion, P. The standard deviation of the sample proportion σ_p is:

  σ_p = sqrt[ P * ( 1 - P ) / n ] * sqrt[ ( N - n ) / ( N - 1 ) ]

  where P is the population proportion, n is the sample size, and N is the population size. When the population size is much larger (at least 20 times larger) than the sample size, the standard deviation can be approximated by:

  σ_p = sqrt[ P * ( 1 - P ) / n ]

• When the true population proportion P is not known, the standard deviation of the sampling distribution cannot be calculated. Under these circumstances, use the standard error. The standard error (SE) can be calculated from the equation below.

  SE_p = sqrt[ p * ( 1 - p ) / n ] * sqrt[ ( N - n ) / ( N - 1 ) ]

  where p is the sample proportion, n is the sample size, and N is the population size. When the population size at least 20 times larger than the sample size, the standard error can be approximated by:

  SE_p = sqrt[ p * ( 1 - p ) / n ]