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Approximate sampling distribution for p, probabilities for p, standard error, standard curve, point estimates, interval estimation

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STAT 301
Brett Hunter

16 September How to approximate a sampling distribution for p Decide on a sample size n Randomly select a sample of size n from the population Compare the proportion of successes, p Repeat steps 2 and 3 ≥ 1000 times. Plot a histogram of the p values to see what the sampling distribution looks like. Properties for the distribution of p Let p be the proportion of successes in a random sample of size n from a population whose proportion of successes is π. Denote the mean value of p’s distribution (thus, of p) by μp, and the standard deviation of p by σp. We know μp = π π(1−π) σp = √ n If it is large and π is not extreme (i.e., π is not close to 0 or 1), then the distribution of p is approximately normal. Rule: Normal if both πn ≥ 10 and (1-π)n ≥ 10 Probabilities for p Say we want to find P(p < .34) We can find this since P~N(μp, σ p) when rule holds. So we standardize p using a Z-score where p−π p−μp π(1−π) Z = σp = √ n And use standard normal tables to solve. Standard error The standard error is an estimate of the standard deviation. The standard error for p, denoted SE(p) is p(1−p) SE(p) = √ n Why do
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