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.
μp = π
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
Z = σp =
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
SE(p) = √ n