Confidence Intervals Summary
A sample of size n is called a simple random sample if it is selected from the population in such a way that every subset of
size n has an equal chance of being chosen as the sample. In particular, every member of the population must have an
equal chance of being included in the sample.
In order to make valid conclusions about a population from a sample, we would like the sample chosen to be
representative of the population as a whole. This means that all the different subgroups present in the population appear
in the sample in similar proportions as they do in the population.
𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑝 =
𝑆𝑎𝑚𝑝𝑙𝑒 𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑝̂ =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒
𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑧𝑒
𝑛𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑠𝑎𝑚𝑝𝑙𝑒 𝑤𝑖𝑡ℎ 𝑎𝑡𝑡𝑟𝑖𝑏𝑢𝑡𝑒
𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑖𝑧𝑒
The population proportion p is a population parameter; its value is constant.
The sample proportion 𝑝̂ is a sample statistic; its value is not constant, but varies from sample to sample.
When the population is large, the sampling distribution of the sample proportion can be determined by assuming that X is
a binomial random variable with parameters n and p. In this case, the mean and standard deviation of the sample
proportion 𝑃̂ are given by:
𝑝(1−𝑝)
𝐸(𝑃̂) = 𝑝 and sd (𝑃̂) = √
𝑛
When the sample size n is large, the sample proportion has an approximately normal distribution with mean 𝜇 = 𝑝 and
𝑝(1−𝑝)
standard deviation 𝜎 = √
𝑛
The value of the sample proportion 𝑝̂ an be used to estimate the population proportion p. Since this is a single-valued
estimate, it is called a point estimate of p.
An interval estimate for the population proportion p is called a confidence interval for p.
An approximate (95% confidence interval for p is given by
𝑝̂ (1 − 𝑝̂ )
𝑝̂ (1 − 𝑝̂
, 𝑝̂ + 1.96√
(𝑝̂ − 1.96√
)
𝑛
𝑛
The distance between the sample estimate and the endpoints of the confidence interval is
called the margin of error (E). For a 95% confidence interval,
𝑝̂ (1 − 𝑝̂ )
𝐸 = 1.96√
𝑛
GENERALLY: