8.3 A Population Proportion

During an election year, we see articles in the newspaper that state confidence intervals in terms of proportions or percentages. For example, a poll for a particular candidate running for president might show that the candidate has 40 percent of the vote within 3 percentage points (if the sample is large enough). Often, election polls are calculated with 95 percent confidence, so the pollsters would be 95 percent confident that the true proportion of voters who favored the candidate would be between 0.37 and 0.43 (0.40 – 0.03, 0.40 + 0.03).

Investors in the stock market are interested in the true proportion of stocks that go up and down each week. Businesses that sell personal computers are interested in the proportion of households in the United States that own personal computers. Confidence intervals can be calculated for the true proportion of stocks that go up or down each week and for the true proportion of households in the United States that own personal computers.

The procedure to find the confidence interval, the sample size, the error bound for a population (EBP), and the confidence level for a proportion is similar to that for the population mean, but the formulas are different.

How do you know you are dealing with a proportion problem? First, the data that you are collecting is categorical, consisting of two categories: Success or Failure, Yes or No. Examples of situations where you are the following trying to estimate the true population proportion are the following: What proportion of the population smoke? What proportion of the population will vote for candidate A? What proportion of the population has a college-level education?

The distribution of the sample proportions (based on samples of size n) is denoted by P′ (read “P prime”).

The central limit theorem for proportions asserts that the sample proportion distribution P′ follows a normal distribution with mean value p, and standard deviation $\sqrt{\frac{p•q}{n}}$, where p is the population proportion and q = 1 -– p.

The confidence interval has the form (p′ – EBP, p′ + EBP). EBP is error bound for the proportion.

p′ = the estimated proportion of successes (p′ is a point estimate for p, the true proportion.)

x = the number of successes

n = the size of the sample

The error bound for a proportion is

$EBP=\left(z_{\frac{\alpha }{2}}\right)\left(\sqrt{\frac{p^{\prime }{q}^{\prime }}{n}}\right)\text{,}$ where q′ = 1 – p′.

This formula is similar to the error bound formula for a mean, except that the "appropriate standard deviation" is different. For a mean, when the population standard deviation is known, the appropriate standard deviation that we use is $\frac{\sigma }{\sqrt{n}}$. For a proportion, the appropriate standard deviation is $\sqrt{\frac{pq}{n}}$.

However, in the error bound formula, we use $\sqrt{\frac{p^{\prime }{q}^{\prime }}{n}}$ as the standard deviation, instead of $\sqrt{\frac{pq}{n}}$.

In the error bound formula, the sample proportions p′ and q′, are estimates of the unknown population proportions p and q. The estimated proportions p′ and q′ are used because p and q are not known. The sample proportions p′ and q′ are calculated from the data: p′ is the estimated proportion of successes, and q′ is the estimated proportion of failures.

The confidence interval can be used only if the number of successes np′ and the number of failures nq′ are both greater than five.

That is, in order to use the formula for confidence intervals for proportions, you need to verify that both $n{p}^{\text{'}}\ge 5$ and $n{q}^{\text{'}}\ge 5$.

Plus-Four Confidence Interval for p

There is a certain amount of error introduced into the process of calculating a confidence interval for a proportion. Because we do not know the true proportion for the population, we are forced to use point estimates to calculate the appropriate standard deviation of the sampling distribution. Studies have shown that the resulting estimation of the standard deviation can be flawed.

Fortunately, there is a simple adjustment that allows us to produce more accurate confidence intervals: We simply pretend that we have four additional observations. Two of these observations are successes, and two are failures. The new sample size, then, is n + 4, and the new count of successes is x + 2.

Computer studies have demonstrated the effectiveness of the plus-four confidence interval for p method. It should be used when the confidence level desired is at least 90 percent and the sample size is at least ten.

Calculating the Sample Size n

If researchers desire a specific margin of error, then they can use the error bound formula to calculate the required sample size.

The margin of error formula for a population proportion is

• $EBP=z_{\frac{\alpha }{2}}\times \sqrt{\frac{p\prime •q\prime }{n}}$, where p′ is the sample proportion, q′ = 1 – p′, and n is the sample size.
• Solving for n gives you an equation for the sample size.
• $n=\frac{{\left(z_{\frac{\alpha }{2}}\right)}^2(p^{\prime }{q}^{\prime })}{EB{P}^2}$. This formula tells us that we can compute the sample size n required for a confidence level of $Cl=1-\alpha$ by taking the square of the critical value $z_{\frac{a}{2}}$, multiplying by the point estimate p′, and by q′ = 1 – p′ and finally dividing the result by the square of the margin of error. Always remember to round up the value of n.