Final answer:
To derive the mean and standard deviation of a proportion for binomially distributed data, you divide the mean and standard deviation of the binomial random variable by the number of trials n, resulting in a mean of p and a standard deviation of √(pq/n) for the proportion.
Step-by-step explanation:
To find the mean and standard deviation of a proportion resulting from n trials with a probability of success p, we use the mean (μ = np) and standard deviation (σ = √npq) of a binomial random variable. When we consider the proportion of successes, the binomial distribution random variable X is divided by the number of trials n to convert it into a proportion, often denoted as P' (the sample proportion).
The mean of the proportion P' is the mean of X divided by n, which gives us μ/n = np/n = p. To find the standard deviation of P', we divide the standard deviation of X by n, resulting in σ/n = √npq/n. This simplifies to σ/n = √(pq/n), as the standard deviation of P'.