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# Confidence Interval for Population Proportion

Confidence Interval for Population Proportion
• Calculus • Decision Maths • Geometry • Mechanics Maths • Probability and Statistics • Pure Maths • Statistics  Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persönlichen Lernstatistiken

Nie wieder prokastinieren mit unseren Lernerinnerungen. 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 Proportion

First, 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?
• Nope! What this does tell you is that “about” $$40\%$$ of them are diseased.

So then, what does “about” mean in technical terms?

Well, it depends on how confident you want to be.

• The confidence interval for the population proportion gives you a range of values near $$40\%$$ that you can say the actual percentage of diseased pods is in.
• The size of the interval will be smaller if you want to be more confident, and it will be larger if you are willing to be less confident.

How can you determine the confidence interval for a population proportion? First, you need to look at some terms you will be using.

### Population Proportion

When 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 Error

The sampling distribution of a statistic has its own standard deviation that describes how much the values of the statistic vary between samples.

• If a sampling distribution is centered closely to the actual value of the population, then a small standard deviation ensures that values of the statistic will cluster tightly around the actual value of the population.

• This means that the value of the statistic will tend to be close to the population value, and you can consider the statistic to be an unbiased estimator of that characteristic.

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,

• $$n$$ is the sample size and

• $$\hat{p}$$ is the population proportion.

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?

Solution:

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 Level

What 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.

 Confidence Level Critical Value $$90\%$$ $$1.645$$ $$95\%$$ $$1.96$$ $$99\%$$ $$2.58$$

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 Error

In 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

1. $$90\%$$,
2. $$95\%$$, and
3. $$99\%$$ confidence levels.

Solution:

Use the critical values at the $$90\%$$, $$95\%$$, and $$99\%$$ confidence levels listed in the table above.

1. For the $$90\%$$ confidence level, the critical value is $$1.645$$, so\begin{align}\text{margin of error for } 90\% &= (\text{critical value})(\text{standard error}) \\&= (1.645)(0.1095) \\&\approx 0.18.\end{align}
2. Similarly, for the $$95\%$$confidence level, the critical value is $$1.96$$, so\begin{align}\text{margin of error for } 95\% &= (\text{critical value})(\text{standard error}) \\&= (1.96)(0.1095) \\&\approx 0.21.\end{align}
3. Finally, for the $$99\%$$confidence level, the critical value is $$1.96$$, so\begin{align}\text{margin of error for } 99\% &= (\text{critical value})(\text{standard error}) \\&= (2.58)(0.1095) \\&\approx 0.54.\end{align}

Let's put the results for the $$95\%$$ confidence level into words.

• Remember that $$40\%$$ of the pods were found to be diseased. What the margin of error tells you is that at the $$95\%$$ confidence level, you can be $$95\%$$ sure that for any random sample taken of the cocoa pods, that the percentage of diseased pods will be $$40\%$$ with a $$21\%$$ margin of error.

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 Proportion

Before determining the confidence interval of a population proportion, two conditions are required to be met by the piece of information given:

1. The data must be representative.

2. The sample size must be large enough.

Let's look at each of those in a little more detail.

### Representative Data

When 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 Size

The 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.

Solution:

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?

Solution:

1. You are trying to find the sample size $$n$$ that gives you both$n \hat{p} \geq 10$and$n (1 - \hat{p}) \geq 10.$
2. Using $$\hat{p} = 0.4$$, that means you need\begin{align}0.4 n &\geq 10 \\n &\geq \frac{10}{0.4} \\n &\geq 25.\end{align}and\begin{align}n (1 - 0.4) &\geq 10 \\n (0.6) &\geq 10 \\n &\geq \frac{10}{0.6} \\n &\geq 17.\end{align}

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 Proportion

To 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,

• $$\hat{p}$$ is the population proportion,

• $$\text{critical value}$$ is the critical value of the confidence level, and

• $$n$$ is the sample size.

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 Proportion

When 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:

1. E: Estimate – Explain what population characteristic you plan to estimate.

2. M: Method – Decide which statistical inference method you want to use.

• To use the confidence intervals for a population proportion method, your problem should meet these requirements:

• The question is asking you for an estimation.

• The situation involves using sample data.

• The type of data involved is one categorical variable.

• There is only one sample.

3. C: Check – There are $$2$$ conditions that must be met in order to use the confidence interval of a population proportion:

• The data must be truly representative of the population and

• The sample size must be large enough.

4. C: Calculate – Use the formula to calculate the confidence interval.

5. C: Communicate – Answer the question asked in the problem, stating what you learned from the data and addressing any potential risks or shortcomings.

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?

Solution:

1. E: Estimate– Explain what population characteristic you plan to estimate.
• You will estimate the value of $$\hat{p}$$, the proportion of cocoa pods on the farm that are diseased.
2. M: Method – Decide which statistical inference method you want to use.

• Because:

• the question is asking you for an estimation,

• the situation involves using sample data,

• the type of data involved is one categorical variable, and

• there is only one sample,

• you can use the confidence intervals for a population proportion method.

• Because a confidence level is associated with the confidence interval, you need to specify a confidence level for the problem. A confidence level isn't given, so use a confidence level of $$95\%$$.

3. C: Check – There are $$2$$ conditions that must be met in order to use the confidence interval of a population proportion:

1. The data must be truly representative of the population.

• It isn't specifically stated how Indodo picked out the pods for the sample, so you don't know that the data is representative.
• That means you will need to assume that the data is representative of the population and include a statement about not having this information when you present the results.
2. The sample size must be large enough.For this example, a success is defined as finding a diseased pod. Since he sampled $$100$$ pods, $$n = 100$$ and\begin{align}\hat{p} &= \frac{ \text{number of successes} }{ \text{sample size} } \\&= \frac{25}{100} = 0.25.\end{align}So,\begin{align}n \hat{p} &= (100)(0.25) \\&= 25 \geq 10\end{align}and\begin{align}n (1 - \hat{p}) &= 100 (1 - 0.25) \\&= 100 (0.75) \\&= 75 \geq 10.\end{align} Because both checks for the required sample size are greater than or equal to $$10$$, the sample size is large enough.

4. C: Calculate– Use the formula to calculate the confidence interval.
1. The sample size is $$n = 100$$.
2. The population proportion is $$\hat{p} = 0.25$$.
3. The confidence level is $$95\%$$, so the critical value of the confidence level is $$\text{critical value} = 1.96$$.
4. Find the standard error.\begin{align}\text{standard error } &= \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{n} } \\&= \sqrt{ \frac{0.25 (1 - 0.25)}{100} } \\&= \sqrt{0.001875} \\&\approx 0.0433\end{align}
5. Find the margin of error.Using the critical value for the $$95\%$$ confidence level, the margin of error is:\begin{align}\text{margin of error for } 95\% &= (\text{critical value})(\text{standard error}) \\&= (1.96) (0.0433) \\&\approx 0.084.\end{align}
6. Find the confidence interval.Now you can construct the confidence interval using:$\hat{p} \pm \text{margin of error} = 0.25 \pm 0.085,$so the interval is:$(0.25 - 0.085, 0.25 + 0.085 ) = (0.165, 0.335).$
5. C: Communicate – Answer the question asked in the problem, stating what you learned from the data and addressing any potential risks or shortcomings.
1. First, a statement about the confidence level.
• The method used to construct the confidence interval will ensure that the actual population proportion is contained in the confidence interval about $$95\%$$ of the time.
• 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.
2. Then, a statement about the results with regard to the actual problem.
• If the sample was selected reasonably, you can be $$95\%$$ confident that the actual proportion of diseased cocoa pods is somewhere between $$0.165$$ and $$0.335$$.
• In terms of percentages, you can be $$95\%$$ confident that the actual percentage of diseased cocoa pods is somewhere between $$16.5\%$$ and $$33.5\%$$.

More examples are always good!

## Examples of Confidence Intervals for a Population Proportion

It 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?

Solution:

1. E: Estimate – Explain what population characteristic you plan to estimate.

• You are estimating the value of $$\hat{p}$$, the proportion of U.S. adults age $$21$$ or older who have moved in with friends or relatives in the past year.

2. M: Method – Decide which statistical inference method you want to use.

• Because:

• the question is asking you for an estimation,

• the situation involves using sample data,

• the type of data involved is one categorical variable, and

• there is only one sample,you can use the confidence intervals for a population proportion method.

• Because a confidence level is associated with the confidence interval, you need to specify a confidence level for the problem. A confidence level isn't given, so use a confidence level of $$95\%$$.

3. C: Check – There are $$2$$ conditions that must be met in order to use the confidence interval of a population proportion:

1. The data must be truly representative of the population.

• It isn't specifically stated how the sample was selected, so you don't know that the data is representative.
• That means you will need to assume that the data is representative of the population and include a statement about not having this information when you present the results.
2. The sample size must be large enough.

• For this example, a success is defined as an adult moving in with friends or relatives. Since the sample included $$843$$ adults, $$n = 843$$ and\begin{align}\hat{p} &= \frac{ \text{number of successes} }{ \text{sample size} } \\&= \frac{62}{843} \approx 0.0735.\end{align}So,\begin{align}n \hat{p} &= (843)(0.0735) \\&= 62 \geq 10\end{align}and\begin{align}n (1 - \hat{p}) &= 843 (1 - 0.0735) \\&= 843 (0.9265) \\&= 781.0395 \geq 10.\end{align} Because both checks for the required sample size are greater than or equal to $$10$$, the sample size is large enough.

4. C: Calculate – Use the formula to calculate the confidence interval.

1. The sample size is $$n = 843$$.
2. The population proportion is $$\hat{p} = 0.0735$$.
3. The confidence level is $$95\%$$, so the critical value of the confidence level is $$\text{critical value} = 1.96$$.
4. The confidence interval is:\begin{align}\hat{p} &\pm (\text{critical value}) \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{n} } \\0.0735 &\pm 1.96 \sqrt{ \frac{(0.0735) (1 - 0.0735)}{843} } \\0.0735 &\pm 1.96 \sqrt{0.00008078} \\0.0735 &\pm 0.0176\end{align} Written in interval notation, you have:\begin{align}\text{confidence interval} &= (0.0735 - 0.0176, 0.0735 + 0.0176) \\&= (0.0559, 0.0911)\end{align}
5. C: Communicate – Answer the question asked in the problem, stating what you learned from the data and addressing any potential risks or shortcomings.

• Confidence interval:If the sample was selected such that it truly represents the population, you can be $$95\%$$ confident that the actual proportion of U.S. adults aged $$21$$ years or older who moved back home or in with friends during the previous year is somewhere between $$0.0559$$ and $$0.0911$$.

• Confidence level:The method you used to determine the interval estimate is successful in capturing the actual value of the population proportion approximately $$95\%$$ of the time.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.

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.

Solution:

1. E: Estimate – Explain what population characteristic you plan to estimate.

• You are estimating the value of $$\hat{p}$$, the proportion of parents between the ages of $$18$$ to $$34$$ years who created a social media account for their babies.

2. M: Method – Decide which statistical inference method you want to use.

• Because:

• the question is asking you for an estimation,

• the situation involves using sample data,

• the type of data involved is one categorical variable, and

• there is only one sample,you can use the confidence intervals for a population proportion method.

• Confidence levels of $$90\%$$, $$95\%$$, and $$99\%$$ were specified.

3. C: Check – There are $$2$$ conditions that must be met in order to use the confidence interval of a population proportion:

1. The data must be truly representative of the population.

• You were told to assume the sample population is representative.
• That means you will need to include a statement about how the results are only valid if the sample selected was representative of the overall population when you present the results.
2. The sample size must be large enough.

• For this example, a success is defined as a parent creating a social media profile for their baby. Since the sample included $$10000$$ parents, $$n = 10000$$ and\begin{align}\hat{p} &= \frac{ \text{number of successes} }{ \text{sample size} } \\&= \frac{(10000)(0.40)}{10000} = 0.4.\end{align}So,\begin{align}n \hat{p} &= (10000)(0.4) \\&= 4000 \geq 10\end{align}and\begin{align}n (1 - \hat{p}) &= 10000 (1 - 0.4) \\&= 10000 (0.6) \\&= 6000 \geq 10.\end{align} Because both checks for the required sample size are greater than or equal to $$10$$, the sample size is large enough.

4. C: Calculate – Use the formula to calculate the confidence interval.

1. The sample size is $$n = 10000$$.
2. The population proportion is $$\hat{p} = 0.4$$.
3. The confidence levels are:
• $$90\%$$, and the critical value of $$90\%$$ is $$\text{critical value} = 1.645$$.
• $$95\%$$, and the critical value of $$95\%$$ is $$\text{critical value} = 1.96$$.
• $$99\%$$, and the critical value of $$99\%$$ is $$\text{critical value} = 2.58$$.
4. The confidence intervals are:
• For a confidence level of $$90\%$$:\begin{align}\hat{p} &\pm (\text{critical value}) \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{n} } \\0.4 &\pm 1.645 \sqrt{ \frac{(0.4) (1 - 0.4)}{10000} } \\0.4 &\pm 1.645 \sqrt{0.000024} \\0.4 &\pm 0.008\end{align} Written in interval notation, you have:\begin{align}\text{confidence interval} &= (0.4 - 0.008, 0.4 + 0.008) \\&= (0.392, 0.408)\end{align}
• For a confidence level of $$95\%$$:\begin{align}\hat{p} &\pm (\text{critical value}) \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{n} } \\0.4 &\pm 1.96 \sqrt{ \frac{(0.4) (1 - 0.4)}{10000} } \\0.4 &\pm 1.96 \sqrt{0.000024} \\0.4 &\pm 0.0096\end{align} Written in interval notation, you have:\begin{align}\text{confidence interval} &= (0.4 - 0.0096, 0.4 + 0.0096) \\&= (0.3904, 0.4096)\end{align}
• For a confidence level of $$99\%$$:\begin{align}\hat{p} &\pm (\text{critical value}) \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{n} } \\0.4 &\pm 2.58 \sqrt{ \frac{(0.4) (1 - 0.4)}{10000} } \\0.4 &\pm 2.58 \sqrt{0.000024} \\0.4 &\pm 0.0126\end{align} Written in interval notation, you have:\begin{align}\text{confidence interval} &= (0.4 - 0.0126, 0.4 + 0.0126) \\&= (0.3874, 0.4126)\end{align}
5. C: Communicate – Answer the question asked in the problem, stating what you learned from the data and addressing any potential risks or shortcomings.

• Confidence interval:Assuming the sample is truly representative of the population, for a $$90\%$$ confidence level, the actual value should be within $$39.2\%$$ and $$40.8\%$$. For a $$95\%$$ confidence level, the actual value should be within $$39.04\%$$ and $$40.96\%$$. Meanwhile, for a $$99\%$$ confidence level, the actual value is expected to fall within $$38.74\%$$ and $$41.26\%$$.What can you make out of the above result, which applies three confidence levels?

• You can tell that the $$90\%$$ confidence level has the smallest range of $$1.6\%$$ (from $$40.8\% - 39.2\%$$), followed by $$95\%$$ with a range of $$2\%$$, and lastly, $$99\%$$ with a range of $$2.6\%$$.

• A smaller interval size means that you are closer to the actual value; however, a lower confidence level means a reduced assurance of the accuracy or precision that the actual value is found there. Now you see, in as much as you wish to have more assurance (as given by the $$99\%$$ confidence level), you would prefer that the interval is as small as possible to narrow the interval closer to the actual value (as given by the $$90\%$$ confidence level). Hence, it is REASONABLY assuring to rely on the $$95\%$$ confidence level.

• Confidence level:The method you used to determine the interval estimate is successful in capturing the actual value of the population proportion approximately $$90\%$$, $$95\%$$, or $$99\%$$ of the time, depending on which interval you choose to consider.You were told to assume the sample was truly representative of the population. Therefore, the results are only valid if the sample selected was actually representative of the overall population.

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 ($$95\%$$).

1. The answer of the first example has an interval size of $$3.52\%$$, while that of the second example has an interval size of $$2\%$$.

What do you think accounts for this differing interval size, although they have the same confidence level of $$95\%$$?

1. You would notice that the first example has a sample size of $$843$$ while that of the second has a size of $$10000$$. It just means that the larger the sample size, the more precise the actual value.

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 $$(0.37, 0.39)$$.

1. What explanation can be given to the difference in the confidence intervals, even within the same area?
2. Whose confidence interval is of higher precision?
3. Assuming both had a $$95\%$$ confidence level, determine which result was derived from a smaller sample size with reasons.
4. Assuming both had used the same sample size, determine who would have used a higher confidence level and give your justification.

Solution:

1. Although both individuals had worked in the same sample area, there are several factors which may affect the uniformity of their results.
1. Firstly, they could have worked with different sample sizes. Remember that a larger sample size would mean a lower margin of error. Hence, the interval difference would be smaller.
2. Another factor is the confidence level. If a higher confidence level is used, even with the same population size, the boundary between the intervals would be wider, meaning less precision. However, a lower confidence level would give a smaller boundary between intervals, meaning more precision but at the expense of the level of assurance.
2. From the intervals, the size of the boundary would be calculated to determine the degree of precision.In Mary's case:$0.41 - 0.34 = 0.07$In Elizabeth's case:$0.39 - 0.37 = 0.02$
• Knowing that a smaller boundary size means a higher precision, you can say that Elizabeth's result is more precise.
3. If both conducted their survey with a $$95\%$$ confidence level, then the sample size becomes the sole basis upon which precision is determined. A larger sample size would mean more precision because of the smaller margin of error. Since Elizabeth's result has more precision, it means she worked with a larger sample size than Mary.
• Therefore, Mary's sample size was smaller.
4. If both conducted their survey with the same sample size, the confidence level becomes the sole basis in determining precision. With Elizabeth's result being more precise, it means a lower confidence limit was used.
• Therefore, Mary would have used a higher confidence level.

Therefore, “the larger your sample size, the more precise you are”.

## Confidence Intervals for a Population Proportion – Key takeaways

• 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.
• $$2$$ major conditions must be met before determining the confidence interval for a population proportion:
1. The data must be truly representative of the population and
2. The sample size needs to be large enough.
• The formula used in finding the confidence interval for a population proportion is:

$\hat{p} ± (\text{critical value}) \sqrt{ \frac{ \hat{p} (1 - \hat{p}) }{n} }$

where,

• $$\hat{p}$$ is the population proportion,

• $$\text{critical value}$$ is the critical value of the confidence level, and

• $$n$$ is the sample size.

• The confidence level varies, but the $$95\%$$ confidence level is more popular among statisticians.
• The steps to follow in finding the confidence interval for a population proportion follow the EMC3steps:
1. E: Estimate – Explain what population characteristic you plan to estimate.
2. M: Method – Decide which statistical inference method you want to use.

• To use the confidence intervals for a population proportion method, your problem should meet these requirements:

• The question is asking you for an estimation.

• The situation involves using sample data.

• The type of data involved is one categorical variable.

• There is only one sample.

3. C: Check – There are $$2$$ conditions that must be met in order to use the confidence interval of a population proportion:

1. The data must be truly representative of the population and

2. The sample size must be large enough.

4. C: Calculate – Use the formula to calculate the confidence interval.
5. C: Communicate – Answer the question asked in the problem, stating what you learned from the data and addressing any potential risks or shortcomings.

To 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.

## Confidence Interval for Population Proportion Quiz - Teste dein Wissen

Question

The confidence interval of a population proportion can be said to be_

the level of certainty that the real or actual population proportion falls within an estimated range of values.

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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.

True.

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Question

True or False?

There are only 3 confidence levels.

False.

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Question

True or False?

Statisticians mostly use the $$90\%$$ confidence level.

False.

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Question

While choosing confidence level, you should tend to be ___.

more precise and more certain.

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Question

True or False?

When determining the confidence interval, you must ensure that the sample data is truly representative of the overall population.

True.

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Question

True or False?

The margin of error depends on your confidence level.

True.

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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?

No.

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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?

Yes.

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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?

Yes.

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Question

True or False?

The confidence interval for a population proportion is the sample proportion plus or minus the margin of error.

True.

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Question

When communicating your result, you must consider these $$2$$ aspects:

The confidence interval and the confidence level.

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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.

True.

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Question

When determining the confidence interval, you must ensure that the sample data is __ of the overall population.

Truly representative.

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Question

True or False?

A $$95\%$$ confidence level would give the same result as a $$90\%$$ confidence level.

False.

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