P utku suleymanoglu umich sampling distributions 17

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Unformatted text preview: ? Sample Proportion Estimator The proportion of individuals with a characteristic in a population, p , is estimated by a sample proportion estimator, which simply calculates the proportion of that characteristic in the sample: C n where C is the number of subjects in the sample which is observed to have the charactestic in question. p= Utku Suleymanoglu (UMich) Sampling Distributions 17 / 21 Sampling Distribution of P ¯ As X , P changes sample to sample, so it is a random variable. Sampling Distribution of P P has a sampling mean of: E (P ) = p and sampling variance: V (P ) = p (1 − p ) n where p is the population proportion. Furthermore, CLT is helpful for sample proportions: As long as np (1 − p ) ≥ 5, you can approximate the distribution of P with a normal distribution with mean p and variance p (1 − p )/n. In class, we talked about how the population values cannot be considered normally distributed in this case, and that they are in fact distributed with a Bernoulli distribution. ¯ This makes C a Binomial(n,p) random variable. Using this we showed how E (P ) and ¯ V (P ) formulas above were derived. ¯ (Why does CLT work here: is P a sam...
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