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Select your languageSuggested languages for you: Confidence Interval for Population Proportion At a cocoa farm, Indodo, the owner of the farm, sampled \(20\) cocoa pods and realized that \(8\) out of those were diseased. In terms of proportion, this means that \(40\%\) of them were diseased! Upon discovering this, Indodo walked out of his farm angrily, thinking that \(40\%\) of all cocoa pods on his farm were diseased. Can you rely on his judgment since he inspected only \(20\) pods out of thousands? What confidence level do you have if you agree that \(40\%\) of all the cocoa pods in Indodo's farm are diseased? Let's take a look at confidence intervals for population proportions, so you can express just how worried about the price of chocolate you should be! The Meaning of Confidence Interval for a Population ProportionFirst, let's take a look at the definition of a confidence interval for a population proportion. A confidence interval for a population proportion can be described as the level of certainty that the real or actual population proportion falls within an estimated range of values. In other words, the confidence interval for a population proportion gives you an estimated boundary or range for which the exact value is expected to be found, with a specified level of assurance. For a reminder about finding these intervals, and the confidence level, take a look at the article Confidence Intervals. Let's go back to the example about cocoa. There were \(20\) cocoa pods sampled and \(8\) out of those were diseased. This gives you a population proportion of \(40\%\). Does that mean \(40\%\) of all the cocoa pods are diseased?
So then, what does “about” mean in technical terms? Well, it depends on how confident you want to be.
How can you determine the confidence interval for a population proportion? First, you need to look at some terms you will be using. Population ProportionWhen it comes to estimating a population characteristic – like population proportion \( (p) \) – your first step is to choose an appropriate sample statistic. What is an appropriate sample statistic to estimate a population proportion? Well, the usual choice is a population proportion, \( \hat{p} \). It is defined by: Population proportion is: \[ \hat{p} = \frac{\text{number of successes}}{\text{sample size}}.\] Let's look at this in an example. In the cocoa example at the start of the article, \(20\) cocoa pods were sampled and \(8\) out of those were diseased. In this context of this example, a success is a pod being diseased. So, \[ \begin{align}\hat{p} &= \frac{\text{number of successes}}{\text{sample size}} \\&= \frac{8}{20} \\ &= 0.4.\end{align}\] Notice that this is the same as the proportion of diseased pods, which is what you would expect. Standard ErrorThe sampling distribution of a statistic has its own standard deviation that describes how much the values of the statistic vary between samples.
For more information about bias, see Sources of Bias in Surveys, Sources of Bias in Experiments, and Biased and Unbiased Point Estimates. Because the standard deviation of a sampling distribution is so important in determining the accuracy of an estimate, it has a special name: standard error. It is defined as: The standard error, \( \sigma \), of a population proportion, \( \hat{p} \), describes how much its values will spread out around the actual value of the population proportion. If the sample size is large, then the standard error tends to be small. The formula for the standard error of a population proportion is: \[ \sigma_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\] where,
In short, an unbiased statistic with a small standard error is likely to result in an estimate that is close to the actual value of the population characteristic. In the cocoa example at the start of the article, \(20\) cocoa pods were sampled and \(8\) out of those were diseased. What is the standard error of the population proportion? Answer: For this example, \(n = 20\) and you have already calculated that \(\hat{p} = 0.4\). Using the formula, \[\begin{align}\sigma_{\hat{p}} & = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\&= \sqrt{\frac{0.4(1-0.4)}{20}} \\&= \sqrt{0.12} \\&= 0.1095\end{align}\] rounded to \(4\) decimal places. Confidence LevelWhat is the confidence level? The confidence level is a measure of the success rate of the method of constructing the interval, not a comment on the population. It is associated with the confidence interval. The confidence level you use can vary, with the popular choices being \(90\%\), \(95\%\), and \(99\%\). The \(95\%\) confidence level is most popular among statisticians because it provides a reasonable compromise between confidence and precision. You may be required to work with \(90\%\) or \(99\%\) confidence levels. This is not an ordeal since it just requires inputting the right critical values. Below is a table of values for \(90\%\), \(95\%\), and \(99\%\) confidence levels.
Be careful here: you can't just use \(0.95\) as the critical value for a \(95\%\) confidence level! This is a common mistake people make. Notice that as the confidence level goes up, the critical value increases. This means that the higher the confidence level you choose, the wider your confidence interval will be. On the other hand, the lower confidence level you choose, the higher risk you run of being incorrect. Margin of ErrorIn the news article example above, the poll results are given as \(65\% \pm 3.2\%\). What's up with the \(\pm 3.2\%\)? That is the margin of error. The margin of error measures the degree of accuracy an estimated result has, as compared to the actual true value, with a certain level of confidence. The margin of error depends on your confidence level, and is also equivalent to half the width of the confidence interval! The margin of error is also related to the standard error, and is the equivalent to the product of the critical value and the standard error. Hence, it is expressed as: \[\text{margin of error } = (\text{critical value})(\text{standard error}).\] Let's go back to the cocoa example. In the cocoa example at the start of the article, \(20\) cocoa pods were sampled and \(8\) out of those were diseased. In the example from the “Standard Error” section, you have found that \( \hat{p} = 0.4 \) and the standard error is about \( 0.1095 \). Find the margin of error for the
Answer: Use the critical values at the \(90\%\), \(95\%\), and \(99\%\) confidence levels listed in the table above.
Let's put the results for the \(95\%\) confidence level into words.
Comparing the results from the three confidence levels, notice that the margin of error goes up as the confidence level goes up! In other words, the more confident you want to be about the result, the larger your error might be. As a warning, the margin of error may not be an accurate one if the sample size isn't large enough! Read on to figure out why. Finding the Confidence Intervals for a Population ProportionBefore determining the confidence interval of a population proportion, two conditions are required to be met by the piece of information given:
Let's look at each of those in a little more detail. Representative DataWhen determining the confidence interval, you must ensure that the sample data is truly representative of the overall population. If this is the case, it is usually mentioned in the problem statement. However, if this is explicitly stated, then you will need to mention it while communicating your findings. In the example about cocoa pods, you don't know how the data is gathered. So, you can't tell whether the data is representative or not. If you do any statistical analysis based on this data, you will need to say something like: No information is given about how the sample was selected. Therefore, the results are only valid if the sample selected was representative of the overall population. When sampling is random, samples can be regarded as representative of the total population. Required Sampling SizeThe sample size must be large enough. This is so you can use the Central Limit Theorem to make the assumption that the distribution is approximately normal. But how do you know how big your sample needs to be? There is a standard check you can do. You need that both: \[n\hat{p}\ge 10\] and \[n(1-\hat{p})\ge 10.\] This condition implies that there are at least \(10\) positive results, as well as a minimum of \(10\) negative results. You may also see the terms 'successes' and 'failures' instead of 'positives' and 'negatives'. Just like most statisticians use the \(95\%\) confidence level, most will also use \(10\) for the number of positive and negative results. You would hope that you actually get a number much larger than \(10\)! In the earlier example with the cocoa pods, is the sample size large enough? On a cocoa farm, Indodo, the owner of the farm, sampled \(20\) cocoa pods and realized that \(8\) out of those were diseased. Determine if the sample size is large enough to find an appropriate confidence interval. Answer: Remember that the population proportion is \( \hat{p} = 0.4 \). Therefore, \[ n \hat{p} = (20)(0.4) = 8 \] and \[ n (1 - \hat{p}) = (20)(0.6) = 12.\] Since \( n \hat{p} < 10 \), this data does not meet the requirements for determining an appropriate confidence interval. That is why the margin of error in the previous example was so large! Let's look at this from a different direction. Assuming that Indodo is doing a random sample, and he finds that \( \hat{p} = 0.4 \) every time, how large does his sample need to be to say that the sample is large enough? Answer:
Choosing the larger of the two values for \(n\), Indodo needs to sample at least \(25\) pods to make sure the sample is large enough to find an appropriate confidence interval. Now that you know when you can find a confidence interval for a population proportion appropriately, let's see how to actually do it. The Formula of Confidence Intervals for a Population ProportionTo determine the confidence interval for a population, use the formula: \[ \hat{p} \pm (\text{critical value}) \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{n} } \] where,
Notice that this is the same thing as: \[ \text{population proportion} \pm \text{margin of error}.\] Rather than using the formula right away, let's look at the steps you would take in actually calculating the confidence interval. Steps in Calculating the Confidence Interval for a Population ProportionWhen you want to find the confidence interval for a population proportion, you follow the \(5\) step process for estimation problems, known by the acronym EMC3. These steps are summarized as:
Continuing with the cocoa farm example: Indodo has decided to do another sample of cocoa pods on his farm. He samples \(100\) of them, and finds that \(25\) of them are diseased. Based on that data, what can you learn about the proportion of cocoa pods that are diseased on the entire farm? Answer:
More examples are always good! Examples of Confidence Intervals for a Population ProportionIt always helps to see the steps used, so let's look at some examples of calculating confidence intervals and discussing the results. You are studying relocation patterns of U.S. adults aged \(21\) years or older who moved back home or in with friends during the previous year. You conducted a survey of \(843\) U.S. adults age \(21\) or older, and \(62\) of them reported that in the previous year they had moved in with friends or relatives. Based on these data, what can you learn about the proportion of all U.S. adults aged \(21\) years or older who moved in with friends or relatives during the previous year? Answer:
Examine your answer and statement of this example you just concluded. You can make comparisons with the findings of your next example. In a study involving \(10,000\) parents, \(40\%\) of parents between the ages of \(18\) to \(34\) years created a social media account for their babies. Assuming this population is representative, determine the confidence intervals for a population proportion with \(90\%\), \(95\%\), and \(99\%\) confidence levels. Answer:
Earlier, you were asked to make comparisons between the results of both examples. The major comparison to be made is the interval size, even with the same confidence level (\(95\%\)).
What do you think accounts for this differing interval size, although they have the same confidence level of \(95\%\)?
Here is another example for more clarity. Mary and her twin sister Elizabeth embarked separately on a random survey in the same area involving the support to build a pilot school. Mary's confidence intervals for the population proportion are \((0.34, 0.41)\), and those of Elizabeth are \((0.37, 0.39)\).
Answer:
Therefore, “the larger your sample size, the more precise you are”. Confidence Intervals for a Population Proportion – Key takeaways
Frequently Asked Questions about Confidence Interval for Population ProportionTo find the confidence interval for a population proportion, choose your confidence level, determine if the sample size is large enough, find the critical value, and then use the formula. This can be described as the level of certainty that the real or actual population proportion falls within an estimated range of values. An example of the confidence interval of population proportion is if a representative population of 200 is sampled and 38% are successes, with a 95% confidence level, the confidence interval of the population proportion is between 31.3% and 44.7%. The formula used in finding the confidence interval for a population proportion is: p'±z'( sqrt(p'(1-p')/n) where p' is the sample proportion, z' is the critical value of confidence level n is the sample size Although both population proportion and population mean are parameters and not statistics, population mean is the average numerical value of a characteristic population, while the population proportion is a fraction of the population exhibiting a characteristic. Final Confidence Interval for Population Proportion Quiz
Question The confidence interval of a population proportion can be said to be_ Show answer Answer the level of certainty that the real or actual population proportion falls within an estimated range of values. Show question
Question True or False? The confidence interval for a population proportion gives you an estimated boundary or range for which the exact value is expected to be found, with a specified level of assurance. Show answer
Question True or False? There are only 3 confidence levels. Show answer
Question True or False? Statisticians mostly use the \(90\%\) confidence level. Show answer
Question While choosing confidence level, you should tend to be ___. Show answer Answer more precise and more certain. Show question
Question True or False? When determining the confidence interval, you must ensure that the sample data is truly representative of the overall population. Show answer
Question True or False? The margin of error depends on your confidence level. Show answer
Question In a population of \(100,000\) people, \(20\) of them were used in a study, and it was observed that \(40\%\) are successes in the study. Is that sample size large enough to determine the margin of error? Show answer
Question In a population of \(100,000\) people, \(200\) of them were used in a study, and it was observed that \(40\%\) are successes in the study. Is that sample size large enough to determine the margin of error? Show answer
Question In finding the confidence interval of a population proportion, Nonny uses: \[ \hat{p} \pm (\text{critical value}) \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{n} } \] Is he correct? Show answer
Question True or False? The confidence interval for a population proportion is the sample proportion plus or minus the margin of error. Show answer
Question When communicating your result, you must consider these \(2\) aspects: Show answer Answer The confidence interval and the confidence level. Show question
Question True or False? The confidence level is a measure of the success rate of the method of constructing the interval, not a comment on the population. Show answer
Question When determining the confidence interval, you must ensure that the sample data is __ of the overall population. Show answer
Question True or False? A \(95\%\) confidence level would give the same result as a \(90\%\) confidence level. Show answer Discover the right content for your subjectsNo need to cheat if you have everything you need to succeed! Packed into one app!Study PlanBe perfectly prepared on time with an individual plan. QuizzesTest your knowledge with gamified quizzes. FlashcardsCreate and find flashcards in record time. NotesCreate beautiful notes faster than ever before. Study SetsHave all your study materials in one place. DocumentsUpload unlimited documents and save them online. Study AnalyticsIdentify your study strength and weaknesses. Weekly GoalsSet individual study goals and earn points reaching them. Smart RemindersStop procrastinating with our study reminders. RewardsEarn points, unlock badges and level up while studying. Magic MarkerCreate flashcards in notes completely automatically. Smart FormattingCreate the most beautiful study materials using our templates. Sign up to highlight and take notes. It’s 100% free. This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept Privacy & Cookies Policy What type of variable is required to construct a confidence interval for a population?The confidence interval for a proportion should be constructed because the variable of interest is an individual's opinion, which is a qualitative variable.
Why would you construct a confidence interval?Why have confidence intervals? Confidence intervals are one way to represent how "good" an estimate is; the larger a 90% confidence interval for a particular estimate, the more caution is required when using the estimate. Confidence intervals are an important reminder of the limitations of the estimates.
What are the conditions for constructing a confidence interval for a proportion?There are three conditions we need to satisfy before we make a one-sample z-interval to estimate a population proportion. We need to satisfy the random, normal, and independence conditions for these confidence intervals to be valid.
What distribution is used to construct confidence intervals with proportions?Standard Normal Distribution Method
The normal distribution can also be used to construct confidence intervals. You used this method when you first learned to construct confidence intervals using the standard error method. Recall the formula you used: 95% Confidence Interval.
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