## Estimating the Proportion of Large Populations

# Statistics Tutorial: Estimating a Proportion (Large Sample)

Suppose we select all possible samples of size *n* from a population of

size *N*. For each sample, we compute a sample proportion, *p*. The

relationship between the sample proportion, *p*, and the population

proportion, π, is described by the sampling distribution of the proportion.

### The Sampling Distribution of a Proportion

When we examine the sampling distribution of the proportion, we find the

following:

- The average of all possible sample proportions is equal to the population

proportion, π. Thus, if there are*k*possible

samples of size*n*, then:π = [ p

_{1}+ p_{2}+ . . . + p_{k}] / k - The standard deviation of the sampling distribution (also known as the

standard error) indicates the “average” deviation between the*k*sample

proportions and the true population proportion, π.

The standard error of the proportion σ_{p}is:σ

_{p}= sqrt[ π *( 1 - π ) / n ] * sqrt[ ( N - n ) / ( N - 1 ) ]where π is the population proportion, n is the sample size, and N is the population size.

- The central limit theorem states that the sampling distribution of any statistic will be normal or nearly normal, if the sample size is large enough. Generally, a

sample size that is greater than or equal to 30 is considered “large enough”.

Therefore, if the sample size is large, the sampling distribution of a

proportion will be approximately normal in shape.

### How to Find the Confidence Interval for a Proportion

A confidence interval provides the most useful estimate of a population proportion. When the sample size is large (greater than or equal to 30), the following six steps can be

used to construct a confidence interval.

- Select a confidence level.
- Compute alpha.
- Identify a sample statistic to serve as a point estimate of the population parameter. Since we are estimating the population proportion, the logical sample statistic is the sample proportion.
- Specify the sampling distribution of the statistic. This distribution (its shape, its mean, and its standard deviation) is described above, in the first part of this

lesson. - Based on the sampling distribution of the statistic, find the value for

which the cumulative probability is 1 – alpha/2. That value is the upper limit of the range of the confidence interval. - In a similar way, find the value for which the cumulative probability is

alpha/2. That value is the lower limit of the range of the confidence interval.

Taken from: *Probability, Statistics, and Survey Sampling **Retrieved November 28, 2006 from http://www.stattrek.com/Lesson4/Proportion.aspx*

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