Unit 2: statistical estimation

2.8 confidence interval for a population proportions

The theory and procedure for determining a point estimator and an interval estimator for a population proportion are quite similar to those described in the previous section. A point estimate for the population proportion is found by dividing the number of success in the sample by the total number sampled. Suppose 100 of 400 sampled said they liked a new cola they tested better than their regular cola. The best estimate of the population proportion favoring the new cola is .25 or 25 percent, found by . Note that a proportion is based on a count of the number of successes relative to the total number sampled.

In case we want to construct confidence interval to estimate a population proportion, we should use the binomial distribution with the mean of population ( ) = n.p where n = number of trials, p = probability of success in any of the trials and population standard deviation = where q is the probability of failure = 1-p. As the sample size increases, the binomial distribution approaches normal distribution, which we can use for our purpose of estimating a population proportion. The mean of the sampling distribution of the proportion of success ( ) is taken as equal to p and the standard deviation for the proportion of successes, also known as the standard error of proportion, is taken as equal to . But when population proportion is unknown, then we can estimate the population parameters by substituting the corresponding sample statistics p and q in the formula for the standard error of proportion to obtain the estimated standard error of the proportion as shown below:

Using the above estimated standard error of proportion we can work out the confidence interval for population proportion as:

Where p = sample proportion of success;
q = 1-p probability of failure
n = number of trials (size of the sample);
Z_/2 = standard variate for given confidence level (as per normal curve area table)
If population is finite, then the standard error of the population proportion is given by
where is the finite population correction factor.
Example: A market research survey in which 64 consumers were contacted states that 64% of all consumers of a certain product were motivated by the product’s advertising. Find the confidence limits for the proportion of consumers motivated by advertising in the population, given a confidence level equal to 0.95.


Solution: - Given
n = 64, p = 64% = 0.64, q = 1-p = 1-.64 = .36
and  = 0.05 = /2 = 0.025 and Z_/2 = 1.96

Thus, 95% confidence interval for the proportion, of consumers motivated by advertising in the population is = .64 1.96 = (.5224, .7576)