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% of the vote within three percentage points (if the sample is large enough). Often, election polls are calculated with 95% confidence, so, the pollsters would be 95% 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, 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 underlying **distribution is a** binomial distribution. (There is no mention of a mean or average.) If *X* is a binomial random variable, then *X* ~ *B*(*n*, *p*) where *n* is the number of trials and *p* is the probability of a success. To form a proportion, take *X*, the random variable for the number of successes and divide it by *n*, the number of trials (or the sample size). The random variable *P′* (read "P prime") is that proportion,

${P}^{\prime}=\frac{X}{n}$

(Sometimes the random variable is denoted as $\widehat{P}$, read "P hat".)

When *n* is large and *p* is not close to zero or one, we can use the normal distribution to approximate the binomial.

$X~N(np,\sqrt{npq})$

If we divide the random variable, the mean, and the standard
deviation by *n*, we get a normal distribution of proportions with *P′*, called the
estimated proportion, as the random variable. (Recall that a proportion as the
number of successes divided by *n*.)

$\frac{X}{n}={P}^{\prime}\text{~}N\left(\frac{np}{n},\frac{\sqrt{npq}}{n}\right)$

Using algebra to simplify : $\frac{\sqrt{npq}}{n}=\sqrt{\frac{pq}{n}}$

** P′ follows a normal distribution for proportions**:
$\frac{X}{n}={P}^{\prime}\text{~}N\left(\frac{np}{n},\frac{\sqrt{npq}}{n}\right)$

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

*p′* = $\frac{x}{n}$

*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**

$EBP=\left({z}_{\frac{\alpha}{2}}\right)\left(\sqrt{\frac{{p}^{\prime}{q}^{\prime}}{n}}\right)$ 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.

### Note

For the normal distribution of proportions, the *z*-score formula is as follows.

If ${P}^{\prime}\text{~}N\left(p,\sqrt{\frac{pq}{n}}\right)$ then the *z*-score formula is $z=\frac{{p}^{\prime}-p}{\sqrt{\frac{pq}{n}}}$

### Example 8.10

Suppose that a market research firm is hired to estimate the percent of adults living in a large city who have cell phones. Five hundred randomly selected adult residents in this city are surveyed to determine whether they have cell phones. Of the 500 people surveyed, 421 responded yes - they own cell phones. Using a 95% confidence level, compute a confidence interval estimate for the true proportion of adult residents of this city who have cell phones.

Suppose 250 randomly selected people are surveyed to determine if they own a tablet. Of the 250 surveyed, 98 reported owning a tablet. Using a 95% confidence level, compute a confidence interval estimate for the true proportion of people who own tablets.

### Example 8.11

For a class project, a political science student at a large university wants to estimate the percent of students who are registered voters. He surveys 500 students and finds that 300 are registered voters. Compute a 90% confidence interval for the true percent of students who are registered voters, and interpret the confidence interval.

A student polls his school to see if students in the school district are for or against the new legislation regarding school uniforms. She surveys 600 students and finds that 480 are against the new legislation.

a. Compute a 90% confidence interval for the true percent of students who are against the new legislation, and interpret the confidence interval.

b. In a sample of 300 students, 68% said they own an iPod and a smart phone. Compute a 97% confidence interval for the true percent of students who own an iPod and a smartphone.

### “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 this method. It should be used when the confidence level desired is at least 90% and the sample size is at least ten.

### Example 8.12

A random sample of 25 statistics students was asked: “Have you smoked a cigarette in the past week?” Six students reported smoking within the past week. Use the “plus-four” method to find a 95% confidence interval for the true proportion of statistics students who smoke.

Out of a random sample of 65 freshmen at State University, 31 students have declared a major. Use the “plus-four” method to find a 96% confidence interval for the true proportion of freshmen at State University who have declared a major.

### Example 8.13

The Berkman Center for Internet & Society at Harvard recently conducted a study analyzing the privacy management habits of teen internet users. In a group of 50 teens, 13 reported having more than 500 friends on Facebook. Use the “plus four” method to find a 90% confidence interval for the true proportion of teens who would report having more than 500 Facebook friends.

The Berkman Center Study referenced in Example 8.13 talked to teens in smaller focus groups, but also interviewed additional teens over the phone. When the study was complete, 588 teens had answered the question about their Facebook friends with 159 saying that they have more than 500 friends. Use the “plus-four” method to find a 90% confidence interval for the true proportion of teens that would report having more than 500 Facebook friends based on this larger sample. Compare the results to those in Example 8.13.

### 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 error bound formula for a population proportion is

- $EBP=\left({z}_{\frac{\alpha}{2}}\right)\left(\sqrt{\frac{{p}^{\prime}{q}^{\prime}}{n}}\right)$
- Solving for
*n*gives you an equation for the sample size. - $n=\frac{{\left({z}_{\frac{\alpha}{2}}\right)}^{2}({p}^{\prime}{q}^{\prime})}{EB{P}^{2}}$

### Example 8.14

Suppose a mobile phone company wants to determine the current percentage of customers aged 50+ who use text messaging on their cell phones. How many customers aged 50+ should the company survey in order to be 90% confident that the estimated (sample) proportion is within three percentage points of the true population proportion of customers aged 50+ who use text messaging on their cell phones.

Suppose an internet marketing company wants to determine the current percentage of customers who click on ads on their smartphones. How many customers should the company survey in order to be 90% confident that the estimated proportion is within five percentage points of the true population proportion of customers who click on ads on their smartphones?