41 A Confidence Interval for A Population Proportion

Throughout an election year, we check out short articles in the newspaper that state confidence intervals in terms of proportions or percentperiods. For example, a poll for a specific candiday running for president could present that the candidate has 40% of the vote within 3 percentage points (if the sample is huge enough). Often, election polls are calculated through 95% confidence, so, the pollsters would be 95% confident that the true propercentage of voters that favored the candidate would certainly be between 0.37 and also 0.43.

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Investors in the stock market are interested in the true proportion of stocks that go up and down each week. Businesses that offer personal computers are interested in the propercentage of family members in the United States that very own personal computers. Confidence intervals have the right to be calculated for the true proportion of stocks that go up or down each week and for the true propercent of households in the United States that very own personal computer systems.

The procedure to find the confidence interval for a populace propercentage is equivalent to that for the population suppose, yet the formulas are a bit various although conceptually the same. While the formulas are various, they are based upon the exact same mathematical structure given to us by the Central Limit Theorem. Thus we will check out the exact same fundamental format making use of the very same 3 pieces of information: the sample value of the parameter in question, the conventional deviation of the pertinent sampling distribution, and also the variety of traditional deviations we must have the confidence in our estimate that we desire.

How perform you understand you are handling a propercentage problem? First, the underlying circulation has actually a binary random variable and also therefore is a binomial circulation. (Tbelow is no point out of a intend or average.) If X is a binomial random variable, then X ~ B(n, p) where n is the variety of trials and also p is the probcapacity of a success. To form a sample propercentage, take X, the random variable for the variety of successes and also divide it by n, the number of trials (or the sample size). The random variable P′ (review “P prime”) is the sample proportion,

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(Sometimes the random variable is dedetailed as

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, check out “P hat”.)

p′ = the estimated proportion of successes or sample proportion of successes (p′ is a suggest estimate for p, the true populace proportion, and hence q is the probcapacity of a faitempt in any kind of one trial.)

x = the number of successes in the sample

n = the dimension of the sample

The formula for the confidence interval for a population proportion adheres to the exact same format as that for an estimate of a populace suppose. Remembering the sampling circulation for the proportion from Chapter 7, the standard deviation was found to be:


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is set according to our wanted degree of confidence and also
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is the traditional deviation of the sampling circulation.

The sample prosections p′ and q′ are approximates of the unrecognized populace proportions p and also q. The approximated proparts p′ and q′ are supplied because p and q are not well-known.

Remember that as p moves better from 0.5 the binomial distribution becomes less symmetrical. Because we are estimating the binomial through the symmetrical normal circulation the even more away from symmetrical the binomial becomes the much less confidence we have in the estimate.

This conclusion have the right to be demonstrated via the following analysis. Proparts are based upon the binomial probcapacity distribution. The feasible outcomes are binary, either “success” or “failure”. This gives increase to a propercentage, interpretation the percentage of the outcomes that are “successes”. It was presented that the binomial distribution can be completely construed if we knew just the probcapacity of a success in any one trial, referred to as p. The expect and also the traditional deviation of the binomial were found to be:


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*

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which is nopoint more than a restatement of the basic standardizing formula via appropriate substitutions for μ and σ from the binomial. We deserve to use the traditional normal distribution, the reason Z is in the equation, because the normal circulation is the limiting distribution of the binomial. This is an additional example of the Central Limit Theorem. We have actually currently watched that the sampling circulation of implies is commonly distributed. Respeak to the extended conversation in Chapter 7 concerning the sampling distribution of prosections and the conclusions of the Central Limit Theorem.

We deserve to currently manipulate this formula in just the exact same way we did for finding the confidence intervals for a expect, however to find the confidence interval for the binomial populace parameter, p.


Wright here p′ = x/n, the suggest estimate of p taken from the sample. Notice that p′ has actually reinserted p in the formula. This is because we execute not recognize p, indeed, this is just what we are trying to estimate.

Unfortunately, there is no correction aspect for instances where the sample size is small so np′ and also nq’ should constantly be better than 5 to develop an interval estimate for p.


Suppose that a industry research firm is hired to estimate the percent of adults living in a big 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 sampled, 421 responded yes – they own cell phones. Using a 95% confidence level, compute a confidence interval estimate for the true propercent of adult inhabitants of this city who have cell phones.


The solution step-by-action.

Let X = the variety of world in the sample that have actually cell phones. X is binomial: the random variable is binary, human being either have a cell phone or they execute not.

To calculate the confidence interval, we must find p′, q′.

n = 500

x = the variety of successes in the sample = 421

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p′ = 0.842 is the sample proportion; this is the allude estimate of the population propercentage.

q′ = 1 – p′ = 1 – 0.842 = 0.158

Since the asked for confidence level is CL = 0.95, then α = 1 – CL = 1 – 0.95 = 0.05

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= 0.025.

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Then

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This have the right to be uncovered making use of the Standard Common probcapacity table in (Figure). This can additionally be uncovered in the students t table at the 0.025 column and infinity levels of freedom because at infinite levels of flexibility the students t distribution becomes the traditional normal circulation, Z.