This document defines key terms related to sampling distributions, including population, parameters, statistics, population proportion, and sample proportion. It explains that as more samples are taken from a population, the sample proportions will begin to form a bell-shaped curve centered around the true population proportion. The mean of all sample proportions equals the population proportion. Under certain conditions where the sample size is large enough and neither the sample proportion or population proportion is too close to 0 or 1, the sampling distribution can be approximated as a normal distribution, allowing us to calculate a z-score to determine how many standard deviations a sample proportion is from the population proportion.

1. Sampling Distribution

2. Vocabulary
•
•
•
•
•
•
Sample – part of a population
Parameters – refer to the population 𝜇, 𝜎, 𝑝
Statistics – refer to the sample 𝑥, 𝑠, 𝑝
Population Proportion = p
Sample Proportion = 𝑝
Sampling Distribution = the distribution of the
sample means or sample proportions

3. Sample proportion vs Population
Proportion
• Suppose p = .64 (population proportion)
• We take 10 samples and find that their sample
proportions are:
• .53, .55, .58, .61, .63, .65, .65, .71, .72, .91
• As we take more and more samples our
sample proportions will begin to make a bellshaped curve centered around the true
population proportion of p=.64

4. Sampling Proportions
• 𝑝=
# 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒
# 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒
= proportion statistic
• 𝑥 𝑝 = the mean of the sample proportions = p
– The mean of all of the sample proportions is equal
to the true population proportion
– We can rarely take all possible samples
– 𝜎𝑝 =
𝑝(1−𝑝)
𝑛
– We can only use this when 10n≤ 𝑁

5. Approximating to Normal Curve
• We can approximate 𝑝 to be Normal if the following
conditions are met
– np≥ 10
– n(1-p) ≥10
– 10n≤ 𝑁
We then can calculate a z-score for a specific sample
proportion against the population proportion
z-score =
𝑝−𝑝
𝜎
This gives the # of standard deviations the sample proportion is
from the population proportion.
You can therefore also find the probability of a sample proportion
occurring from the z-score