Confidence Intervals for Population Proportion - Online Tutor, Practice Problems & Exam Prep

Constructing Confidence Intervals for Proportions

Earlier in this chapter, we learned about confidence intervals, and we know how to find their associated critical z-scores. But we have yet to fully construct a confidence interval for a population proportion p, so that's exactly what we're going to do here, putting everything that we've learned so far together in a step-by-step process to ultimately construct a confidence interval for p. So let's go ahead and get started here by jumping right into our example. Here, we're told that from a survey of 200 people, 90 of those people preferred computers from brand A over brand B and we're asked here to construct a 90% confidence interval for the true proportion of people who prefer computers from brand A. Now remember that saying the true proportion is really just referring to the population proportion, p.

Now in this problem, since we're given the number of people that preferred computers from brand A and the total number of people that were surveyed, we can easily divide that 90 by that 200 to get p̂, which is what we want to use as our point estimate here. So to construct our confidence interval for p, we want to use that point estimate p̂, along with our margin of error, which we can find using this equation. That e, our margin of error, is equal to that critical z-value, that's z_α/2, times the square root of p̂ (that point estimate) times (1 minus p̂) over n, our total sample size. So that's how we can ultimately construct a confidence interval for our population proportion. But before we can fully calculate that margin of error to construct that confidence interval, we need to go ahead and start from step 1 and verify that we can even use this process.

Remember that the sampling distribution of p̂, which we know is x over n like we talked about earlier, we know that the sampling distribution is approximately normal when both np and nq are greater than or equal to 5. Now, if we work this out mathematically, all this really means is that we have at least 5 successes and at least 5 failures. So that's all we need to verify here. Now looking at the specific problem that we're working with here, we were told that 90 people preferred brand A. So those are successes here.

Now that's definitely at least 5, so we can go ahead and check that off. Now since we surveyed 200 people in total, that means that 110 of them preferred brand B, representing a failure in this specific scenario. So that's definitely also at least 5. We can go ahead and move on to step 2 here, finding our critical z-value, z_α/2. Remember that this is entirely based on what our confidence level is.

So with this 90% confidence level, we know that α/2 is going to be equal to 1 minus 0.9 (that confidence level), over 2, which gives us 0.05. Now using either a table or a calculator, this ultimately gives us that our critical z-value, z_α/2, is equal to 1.645. So with that critical z-value found, we can go ahead and move on to step 3. Now in step 3, this is specifically if this is not already given to us. Here, we want to find p̂, our point estimate, which is equal to x over n.

Now in this case, p̂ is not given to us explicitly, so we do need to go ahead and calculate it by taking x and dividing it by n. So taking that x value of 90 and dividing it by my total sample size of 200, this gives me p̂ equals 0.45. So with that point estimate found, we now need to determine our margin of error using the equation that we looked at above. Remember that our margin of error is going to be equal to that critical z-value that we just found times the square root of p̂ (that point estimate) times (1 minus p̂) over n. So looking at this margin of error here, e is going to be equal to 1.6451.645√0.45(1-0.45)200times the square root of 0.45 (that point estimate), times 1 minus 0.45 (which is 0.55) over that total sample size, which in this case is 200.

Now working out this algebra here, multiplying all of this together ultimately gives us a margin of error of 0.0579. Now that we have that margin of error, we can move on to our final step in actually finding our confidence interval by subtracting and adding that margin of error to our point estimate, p̂. So constructing this confidence interval here, I'm going to go ahead and abbreviate that as c I. I'm going to take my point estimate, p̂, that's that 0.45, and go ahead and subtract that 0.0579, that margin of error that we just found for that lower bound, and then go ahead and add 0.45 plus 0.0579 for that upper bound. Now this ultimately gives me a lower bound here of 0.3921 and an upper bound of 0.5079.

Now we've constructed our confidence interval. But what exactly do these values mean? Often, the most important part of constructing our confidence interval is being able to interpret it. So going back to our original problem here, we know that we are constructing a 90% confidence interval. So that means that based on this confidence interval that we found, we can say that we are 90% confident that the true proportion of people who prefer brand A Computers is in between these two values.

So as a percentage, that's 39.21% and 50.79%. So now we've successfully constructed and interpreted this confidence interval. Now we're going to continue getting a bit more practice with this coming up next. Let us know if you have any questions, and I'll see you in the next one.

Over the first $20$ days of the semester, one student is late to class on $6$ days. Find the margin of error for a $98\%$ confidence interval for the true proportion of time this student is late.

• A. $E=0.3$
• B. $E=0.238$
• C. $E=0.169$
• D. $E=0.062$

Over the first $20$ days of the semester, one student is late to class on $6$ days. Construct a $98\%$ confidence interval for the true proportion of time this student is late. 

• A. (0.276, 0.324)
• B. (0.131, 0.469)
• C. (0.3, 0.7)
• D. (0.062,0.538)

Constructing Confidence Intervals for Proportions Example 1

If you haven't already given this problem a try on your own, go ahead and do that before checking back in with me. Alright. In this problem, we're told that Apple conducted a survey to see what percent of people preferred the iPhone over all other types of smartphones and they found that 62% of the 100 sampled individuals preferred the iPhone. We're asked to do a couple of different things here. The first thing we want to do is construct a 99% confidence interval.

Then we also want to construct a 92% confidence interval for the proportion of individuals who prefer the iPhone. Then we have a question to answer based on that, but let's go ahead and get started with these confidence intervals. Essentially, we just need to follow these steps two times because we're creating 2 confidence intervals. So starting with step 1 here, we want to verify the number of successes and failures are at least 5. Now because 62 percent of our 100 individuals prefer the iPhone, 62 out of 100, that's our 62%, is definitely greater than or equal to 5.

Then the remaining 38 that did not prefer the iPhone, that is still also greater than or equal to 5. Now this will hold true for both of our confidence intervals because we're working with the exact same scenario, just different levels of confidence. So moving on to step 2 here, we want to find our critical value, that's z alpha over 2. So starting with part a, our confidence level in this case is 99%. So to find z alpha over 2 alpha over 2 is going to be 1 minus that confidence level 0.99 over 2.

This gives us here a value of 0.005, which then corresponds to a critical z score, z alpha over 2 of 2.576. So with that z value found, the next thing we want to do is find p hat, which is our x over n. Now, in this case, we were already actually given a p hat, so we don't have to calculate it. P hat is equal to 0.62. That's that 62%.

So now we can find our margin of error. Now our margin of error is going to be equal to this equation, z alpha over 2 times the square root of p hat times 1 minus p hat over n. So finding this margin of error here, e is equal to 2.576 times the square root of p hat. That's 0.62 times 1 minus p hat. That's going to be 0.38 over n, our total number of sampled individuals, which in this case is 100.

So putting that 100 in there, then doing this algebra multiplying all of this out, if we type it into a calculator, we will get a margin of error here of 0.125. So we can actually construct our confidence interval by now subtracting and adding this margin of error to our point estimate, p hat. So our confidence interval here, this is our 99% confidence interval based on that Z Alpha over to our critical value. So this is going to be p hat. That's 0.62-0.125 for that lower bound and 0.62 plus 0.125 for that upper bound.

Now working this out, this means that that lower bound is equal to 0.495 and 0.745. So this is our confidence interval. This is our 99% confidence interval. Now looking at the math that we did here, the only thing that needs to change in part b where we want to construct a 92% confidence interval is this z alpha over 2 value. That's the only thing that's based on our confidence level.

So we can perform almost the exact same calculation, but first we need to determine our critical value. So for this 92% confidence interval, our critical value corresponds to alpha over 2 being equal to 1 minus 0.92 over 2. That gives us 0.04, which then corresponds to a critical Z value of 1.751. So all we need to do now is plug this 1.751 in where this 2.576 was originally, keeping everything else the same. So calculating this margin of error, e, will end up giving us 1.751 times the square root of 0.62 times 0.38, just copying this exactly as it was above, over 100.

Working this out gives us a margin of error of 0.085. So we can already see that we have a smaller margin of error here for this 92% confidence interval. So actually constructing this confidence interval p hat is the exact same. It is still that 0.62. So this is 0.62-0.085 and 0.62 plus 0.085.

Now if we actually work this out, this then means that our confidence interval, that lower bound is 0.535 and our upper bound is 0.705. So we now have both of our confidence intervals here. Now, our last question asks us here, this is part c, why are these two confidence intervals different, even though they were based on the same sample information? So let's think about this here. The only thing that was different is the confidence level between these two confidence intervals.

For this first one, this was a 99% confidence interval. So we would say that we are 99% confident that the true proportion of people that prefer iPhones is between 49.5 percent and 74.5 percent. Whereas for our 92% confidence interval, we would say that we are 92% sure that the true proportion of people that prefers iPhones over all other smartphones is between 53.5 percent and 70.5%. So we can see that this is a smaller range of values than it was for our 99% confidence interval. Now the reason for this is that level of confidence.

Because this is a larger range, we can say that we are more confident that the true proportion will actually be in that range. Whereas if we're only 92% confident, we can have a smaller range of values. Now feel free to let us know if you have any questions here, and let's keep practicing.

Constructing Confidence Intervals for Proportions Example 2

If you haven't already tried to solve this problem on your own, go ahead and give it a try before checking back in with me. Here, we're told that for a fair pair of dice, the probability of getting snake eyes, that's two ones, is 1 out of 36. Now you have a pair of dice that could be weighted to ones more often. Now you roll the dice 50 times and get snake eyes on 11 of those rolls. We want to determine a couple of different things here.

The first is that we want to create a 99% confidence interval for the true proportion of snake eye rolls. Then we want to determine if 1 over 36, the probability that was stated at the beginning, is in this confidence interval that we created. Then, based on that, are the dice weighted? So starting with part a here, we want to go ahead and follow our steps to construct a confidence interval. Starting from step 1 here, we want to verify the number of successes and failures are both greater than or equal to 5.

Because we got snake eyes on 11 of those rolls, that is 11 successes. Out of a total of 50, that means that our failures were also at least 5. So both of these are true. We can move on to step 2, finding our critical value, z_α/2. So in order to find this, we are looking at that confidence level.

So if we calculate α/2, that's going to be 1 minus 0.99 over 2, which gives us a value here of 0.005. Now, if we use a z table or a calculator, we will end up finding that this corresponds to a critical z score, z_α/2 of 2.576. So with that critical value found, we can move on to step 3. If it is not given to us, this is when we want to find p̂, which is equal to x / n. Now we are not explicitly given p̂, so we do need to go ahead and calculate it.

In our problem, we have x being equal to 11 and a total number of trials of 50. So p̂ is equal to that 11 over 50, that's x / n, which ends up giving us here 0.22. So we have that point estimate p̂. Now we can move on to step 4, that's finding our margin of error using this equation here which we're going to plug in the values that we've already found to this margin of error equation. So our margin of error is going to be equal to 2.576 times the square root of (p̂ (0.22) times (1 - p̂) (0.78) over n, in this case, 50.

Now if we work out this math, type this into our calculator, we'll get here 0.151. So now moving on to step 5, actually constructing our confidence interval, we're going to subtract and add our margin of error to our point estimate. So taking our confidence interval here, that's going to be 0.22 (our point estimate, p̂) minus 0.151 and 0.22 plus 0.151. So this results in a lower bound here of 0.069 and an upper bound of 0.371.

So if we think about these in terms of percentages, we can move our decimal point over, and this gives us 6.9% and 37.1%. So, based on what this confidence interval actually means with our confidence level of 99%, we can say that we are 99% confident that the true proportion of snake eye rolls happens between 6.9% and 37.1% of rolls. So, based on these values, we can move on to part b. We want to determine if 1 over 36 is in this confidence interval that we created. Now one over 36 as a decimal is equal to 0.028.

So as a percentage, that is 2.8%. We can see that that does not fall within our confidence interval. So, the answer to part b is no, that is not in the interval. So based on that, we move on to part c. Are the dice weighted?

Even though we can't say with 100% confidence, we can say that we are pretty sure that these dice are weighted because this 2.8% of the time is much less than that lower bound of our interval. It's not within our confidence interval at all. So we can say with a lot of certainty that these dice are weighted. If you have questions, feel free to let us know. I'll see you in the next one.

Finding the Minimum Sample Size Needed for a Confidence Interval

When constructing a confidence interval for a population proportion, we are often given our sample size n, and we can plug it into this equation to find our margin of error and ultimately construct our confidence interval. But in some cases, we won't actually be given our sample size, and instead, we'll have to find out what it needs to be to result in some given margin of error. Now in order to do this, we can just take the margin of error equation that we already know and rearrange it for n, our sample size. We can then plug in values to ultimately figure out what n needs to be. Now we're going to work through a problem together here doing just that.

So let's go ahead and get started by jumping right into our example. Here, we're told that you are running for student body president and you want to estimate the proportion of people who would vote for you with 90% confidence. Now you also want your margin of error to be 0.02 or smaller, and we want to find here how many students are needed in your sample in order to get this margin of error. So instead of being given our sample size and needing to calculate our margin of error to construct our confidence interval, we are instead given our margin of error, in this case, 0.02, and asked to find how many students are needed in your sample.

That's our sample size n. Now we know that with anything, we want our margin of error to be small and having a larger sample size results in a smaller margin of error. So here, we want to determine the minimum sample size needed in order to get this small margin of error, in this case of 0.02. Now as I mentioned earlier, we can take that margin of error equation that we know and rearrange it for n. So doing that here gives us that n, our sample size, is going to be equal to our critical z value, that's z_α/2, over that margin of error e, and that whole term is squared.

Then it gets multiplied by p̂ times 1 minus p̂. Now having rearranged this margin of error equation for n, we can then just plug in values that are given to us in our problem. And then once we get an answer, if it is a decimal, we're going to want to round up to the next whole number because we know that we can't just sample part of something. So let's go ahead and come back down to our problem here and look at all of the values that we're given. We have our margin of error of 0.02 and then based on our 90% confidence level, we know that this corresponds to a critical z value, z_α/2, of 1.645.

We can find that by either looking at a table or using our calculator. Now we're not actually given a p̂ in this problem, and we need p̂ to find n, our sample size. Whenever this happens, whenever you're not given p̂, you want to go ahead and use a p̂ of 0.5. This is because using p̂ equals 0.5 is going to give you the highest necessary sample size to get that margin of error that you're working with no matter what p̂ actually is. So with all of these values, let's go ahead and plug them into our equation here.

So our sample size, n, is going to be equal to that critical z value, 1.645, over that margin of error, that 0.02, that is squared. And then since we were not given p̂, we are using p̂ equals 0.5, so we want to multiply by 0.5, that's p̂, and then 1 minus p̂ is also 0.5. Now actually multiplying all of this out gives us a value of 1,691.27. Now remember, since we're given a decimal here, we want to go ahead and round this up to the next whole number because we can't be working with 0.27 of a student in this survey. So this then results in a value of 1,692 students.

That's how many are needed in our sample in order to get this margin of error. So this is our final answer here. That's our minimum sample size needed. Now we're going to continue getting practice with these types of problems coming up next, so feel free to ask us if you have any questions. I'll see you in the next video.

A previous study found that your school consists of $60\%$ White/Caucasian students. You want the $98\%$ confidence interval for the proportion of White/Caucasian students to be no more than $.05$ away from the true proportion. How many students must you include in a sample to create this confidence interval? 

• A. 520
• B. 519
• C. 542
• D. 260

You wish to estimate with $90\%$ confidence the population proportion of people in Gen-Z that use social media. If you want your estimate to be accurate within $4\%$ of the population proportion, what is the minimum sample size needed? 

• A. 422
• B. 423
• C. 405
• D. 406

Your company has asked you to estimate the proportion of people who prefer the color red over other primary colors for manufacturing purposes. If they want the estimate to be within $.01$ of the true proportion with $95\%$ confidence, how many people should you survey?  

• A. 1825
• B. 6766
• C. 9604
• D. 97

You want to make a $97\%$ confidence interval for the population proportion of people between $20-30$ years old who have gotten a speeding ticket in the past $2$ years. A prior study found that $26\%$ of people between $20-30$ years old have received a speeding ticket in the last year. If you want your estimate to be accurate within $3\%$ of the true population proportion, what is the minimum sample size needed? 

• A. 1007
• B. 1006
• C. 579
• D. 1309