P′ = the estimated proportion of successes or sample proportion of successes ( p′ is a point estimate for p, the true population proportion, and thus q is the probability of a failure in any one trial.) (Sometimes the random variable is denoted as P ^ P ^, read "P hat".)
The random variable P′ (read "P prime") is the sample proportion, To form a sample proportion, take X, the random variable for the number of successes and divide it by n, the number of trials (or the sample size). (There is no mention of a mean or average.) If X is a binomial random variable, then X ~ B( n, p) where n is the number of trials and p is the probability of a success. How do you know you are dealing with a proportion problem? First, the underlying distribution has a binary random variable and therefore is a binomial distribution. Because of this we will see the same basic format using the same three pieces of information: the sample value of the parameter in question, the standard deviation of the relevant sampling distribution, and the number of standard deviations we need to have the confidence in our estimate that we desire. While the formulas are different, they are based upon the same mathematical foundation given to us by the Central Limit Theorem. The procedure to find the confidence interval for a population proportion is similar to that for the population mean, but the formulas are a bit different although conceptually identical. Confidence intervals can be calculated for the true proportion of stocks that go up or down each week and for the true proportion of households in the United States that own personal computers. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. Investors in the stock market are interested in the true proportion of stocks that go up and down each week. Often, election polls are calculated with 95% confidence, so, the pollsters would be 95% confident that the true proportion of voters who favored the candidate would be between 0.37 and 0.43. For example, a poll for a particular candidate running for president might show that the candidate has 40% of the vote within three percentage points (if the sample is large enough). During an election year, we see articles in the newspaper that state confidence intervals in terms of proportions or percentages.
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