Two Proportions: Videos & Practice Problems

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Conducting a hypothesis test for two sample proportions involves comparing the difference between them. Begin by formulating the null hypothesis, $H$<sub>0</sub>, stating $p_1=p_2$. Calculate the z-score using the difference in sample proportions and pooled variance. The p-value determines if the null hypothesis is rejected. For confidence intervals, use the point estimate and margin of error to assess the difference. Ensure conditions like normal distribution and independence are met for valid results.

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Difference in Proportions: Hypothesis Tests

Welcome back, everyone. So we spent a lot of time learning how to conduct a hypothesis test of one sample when we're doing a proportion. But as you get deeper into the course, some of the problems you may start to run into will give you two sets of data or two samples and ask you to conduct a hypothesis test where you're looking at the difference between these two proportions. Now this might seem like it's going to be twice as complicated or twice the amount of work, but, thankfully, all of the basic steps are exactly the same. We're going to write some hypotheses, calculate some test statistics, which are z scores, find a p-value, and then write a conclusion.

Alright? Now, of course, there are a few things that are a little bit different, but don't worry because I'm going to walk you through all of this. We're going to jump right into an example, and I'll show you how it works. Alright? Let's get started.

So, basically, the basic difference here is that in a hypothesis test with two samples instead of one, we're going to test claims about the difference in the two proportions. So let's go ahead and just take a look at our problem over here. So this table that's shown to us on the right summarizes a study done on the success rate, so there are percentages, of a nicotine patch in helping people quit smoking. So we're going to do a hypothesis test within a significance level alpha as 0.05 to determine if the proportion in these two subjects is different in the two groups. This is always how these problems will go.

They'll give you two samples and ask you to do a hypothesis test about the difference in the groups, and we're doing a proportion here. Alright? So remember, basically, all we're going to do is going to go through the steps, but there are a few conditions that we want to check first. Let's just go through them really quickly here. So we're just going to check that these samples are random at independence.

Almost always, you can assume that these things are. We're going to grab some random subjects, both, you know, which don't interfere with each other, and they're randomly sampled. Now we just have to check if there's five successes and failures in each sample so that they're normally distributed. Here we have, 11 successfully quit out of twenty and seventeen out of 23. So there's greater than five successes and failures in both.

So this is a pretty quick thing to check off really quickly at the very beginning of the problem. Alright? So the first step here is we have to write our initial hypotheses, our H0 and our H1. Now how does this work? Well, basically, what we had is when we had for one proportion over here, we always had that p was equal to some number that we pulled out of the problem.

It was 40% or 0.65 or something like that. But now that we're taking a look at two samples over here and we're testing claims about the difference between the two proportions, what you're always going to do in these problems is you're going to write your H0, your initial hypothesis, as p₁ = p₂. Your default assumption is that there's no difference in the proportions and they're exactly the same. Another way of writing this, by the way, is that the difference in proportions is actually just zero. \( p_1 - p_2 = 0 \).

We're going to see some advantages of writing it this way in just a second here. Alright? So basically, our initial hypothesis, our null hypothesis, is that p₁ is equal to p₂, that the proportions of people who quit using these two things are exactly the same. Alright? Now what about the alternative hypothesis?

Here's where we're going to have to use either a less than, greater than, or not equal to sign. This works exactly the same. We're going to have to figure out what it is based on the wording of the problem here. Now inside of this problem here, there's no indication that we're looking for, you know, evidence that the proportion is less than or greater than. We don't see any of those words.

So we're basically just going to use a not equal sign like this. Alright? So by the way, this just means this is where we're going to use a two-tailed test, and this is going to be useful later on when we look at the distribution. Alright? Okay.

So now we're finished off with step one. We're going to move on to finding out what our test statistic is. Now for one proportion, we calculated a z score using this equation for two samples. We're going to use something that's a little bit more complicated, but, basically, it's actually kind of very similar. We always had a sample proportion, minus a parameter.

In this case, it's the same except this is a difference in sample proportion minus a difference in the parameter. And then we always had a square root thing on the bottom, which had some standard deviations, and we'll talk about that in just a second here. Alright? Now a lot of these problems will have many different numbers flying around or lots of numbers just indicated in different tables. It's always a good idea to get really organized with this, and figure out what your two groups of data are going to be.

So in this case, what I've done here is I've labeled my first group as the placebo and the second as the patch, and I'm just basically going to fill out my n's, x's, and p's, just using the data that's given to me already. So for the first one, they have 20 subjects and 11 who successfully quit. And then the second one, for the patch, I had 23 total subjects, and I had 17 who quit. That's just successes and the sample sizes. Alright?

Now, again, we're going to use if you take a look at the first, item that's in their first term that's in this z score equation, we're going to have to take a look at the difference in the sample proportions. Now in order to calculate those, I can do that really quickly with the four numbers that I've just used. I've just written over here. This is just going to be \( \frac{11}{20} \), which is 0.55. You just take x over n.

And then for a p₂, you're just going to do \( x \) over \( n \) \( \frac{17}{23} \), which ends up being 0.74. Alright? So that's basically just your p₁ minus p₂. So that's the first term that goes inside of this equation over here. So we have a parenthesis, we've got 0.55 minus 0.74.

A human resources department is comparing two employee training programs to see if they lead to different pass rates on a required certification exam. They randomly select two groups of employees. In Program A, 16 out of 20 employees passed the exam. In Program B, 30 out of 40 employees passed. Are the basic conditions met to conduct a 2-proportion hypothesis test?

• A. Yes, the basic conditions are met.
• B. No, the basic conditions are not met.
• C. There is not enough information to answer the question.

Answer: (B)

A school administrator wants to compare the proportion of students who passed a standardized math exam in two different schools by taking samples from 2 classes. Assume the samples are random and independent. Calculate the $z$-score for testing whether there is a significant difference in the population proportions of student passing rates, but do not find a $P$-value or draw a conclusion for the hypothesis test.


Class A: 72 out of 120 students passed.
Class B: 65 out of 100 students passed.

• A. -0.76
• B. 1.07
• C. -1.07
• D. 0.76

Answer: (A)

Difference in Proportions: Hypothesis Tests Example 1

Welcome back, everyone. So in this example, we're given some information about these two samples of data, and we're asked to conduct an entire hypothesis test from start to finish. Now bear with me here because, remember, there are lots of steps in a hypothesis test, and there are many different calculations to do. So it's going to take us a little bit of time to get through this problem, but just bear with me. We're going to go through all of the steps, and we're going to get through it together.

Alright. Let's check it out here. So, a study on the effectiveness of seat belts in reducing injuries is done on two random samples of drivers. Now, we get information among the group who are not wearing a seat belt; we're told that fifty out of fifty are injured and two thousand three hundred fifty are not. Now, among a second group who were wearing a seat belt, fifteen were injured and fifteen eighty-five were not.

So, we're going to use a 0.01 significance level, remember that's your alpha, to test the claim that not wearing a seat belt results in a greater proportion of injuries. Okay? Now, remember, the first thing is you always just want to go through a couple of the conditions real quick. So, we're basically just going to check if the samples are random and independent. Sometimes you can assume it.

In this case, we're actually just told that these are random samples, and you can assume that they're independent because you have drivers who were not wearing a seat belt and that doesn't affect the ones who were. So, we have random and independence there. We have the two sets of data here. We have fifty and two thousand three hundred fifty, and then fifteen and fifteen eighty-five. So basically, there are greater than five successes and failures in each of them.

Alright? So those conditions are out of the way. Now remember, the first thing you have to do here with any hypothesis test is always you want to write out what your null and alternative hypotheses are. So basically, this is just going to be your H₀ and your H₁ for a two-proportion hypothesis test. Remember, for H₀, it's always just that the default assumption is that the two proportions are equal to each other.

We're testing a claim that basically, there's a difference in the proportion, so your default assumption is that there is no difference, is that p₁ is equal to p₂. Alright? Now here's where it gets a little bit tricky. Right? Because you have your comparison of p₁ and p₂, and you're going to have a sign that goes here. It's going to be less than, greater than, or not equal to. Now before we do that, let's just actually figure out what is group one and group two. Alright. So we're told here we have a group who are not wearing a seat belt, and we have some information, and then group who were wearing a seat belt. So basically, what I'm going to do is I'm just going to call this group one, and then the people who were wearing a seat belt, I'm going to call that group two.

Alright? So just so I can actually find what my group one is and group two, that's going to affect my p₁ and p₂ and so on and so forth. Alright? So in other words, what happens here is we're going to have that in the last part of the sentence here, we're trying to test the claim that not wearing a seat belt, that's group one, results in a greater proportion of injuries. So in other words, we're trying to test the claim here that p₁ is actually greater than p₂.

Alright? That's the sign that goes there. Okay? So we have p₁ is greater than p₂ and our next step is to move on to calculating our test statistic, which is going to be a z-score.

Now remember, you should always start to organize your information over here. So remember, we had these two groups of people. We had the ones who were not wearing a seat belt, so no seat belts. That's going to be the group over here, and then the ones who were wearing a seat belt. Alright?

So that's going to be group two. Okay. We're told over here that fifty drivers were injured. Remember, we're going to call that a success even though that it's not necessarily something that is good, but that's going to be our success, and then we have two thousand three hundred and fifty were not. Remember, we have x₁ and n₁.

So x₁ in this case is just going to be 50. But what is n₁? Is this just the 2,350 that's given to us? Well, be really, really careful here. It's actually not. A lot of problems will try to trick you. Basically, they'll give you two numbers, and then you're just going to assume that you're just going to write those down as x₁ and n₁. What happens here is that fifty drivers are injured and two thousand three hundred fifty were not injured. So that means that the total is not two thousand three hundred fifty; it's the combined it's the combination of the twenty-three fifty and fifty.

Alright? So it's 2,350 plus 50. That ends up being your total, which is 2,400. Alright? So that's how many trials were in that first group.

Alright? So just be really, really careful here. I'm actually just going to write a little bit of notes, just so you can always just kind of watch out for this because some problems will try to trip you up this way. Alright?

Difference in Proportions: Confidence Intervals

Welcome back, everyone. So in the last couple of videos, we've seen how to conduct a hypothesis test of proportions using two samples instead of one. Now in some problems like the one we're going to work out down here, you may also be asked to construct or build a confidence interval for the difference in proportion. We've already seen all of the steps to building a confidence interval when it came to one sample proportions. And fortunately, all of the steps are exactly the same when it comes to a difference in proportions.

There's really just a couple of tiny things that we'll need adjustments for when it comes to the point estimator and margin of error equations. We'll see a slightly modified equation for e. Otherwise, everything is exactly the same. So let's jump right into this problem over here.

It's one we've seen the numbers for already. Let's get started. Again, the basic difference for these types of problems with two samples is that we're going to be making a confidence interval for the difference in the population proportions by using an estimator using the sample proportions and then a margin of error. That's going to be slightly different.

So we have a table that summarizes the study of the success rate as a percentage of a nicotine patch in helping people to quit smoking. We're going to make a 90% confidence interval for the difference in success between the two groups.

Now, the second part of the problem we're going to get to at the very end of the video, I promise I'll get back to it, but let's just jump right into the steps here because we already know how to do a lot of this stuff. So for the first step, we're just going to verify a couple of really quick conditions for each sample. We just need to have more than five successes and failures so we can assume that it's normally distributed.

As we've seen in our previous video, those conditions apply. So let's jump into the second step, which is going to find the critical z-value, which comes directly from your confidence level, has nothing to do with the sample data. Our confidence level in this problem is 90%, or as a decimal point 0.90.

This is going to tell you what your critical z-value is, basically the amounts that's on both sides of the tails of the distribution, and that z score over here is going to be 1.645. We've seen that number over again.

Now that we have our critical value and we've verified the conditions, we're going to go ahead and build our confidence interval.

The first thing we need here is we need a point estimate, basically, our best guess. It also forms the middle number of our confidence level or our confidence interval. This is basically just going to be a difference in the sample proportion. Instead of using one proportion, we're just going to use the difference in the two samples.

So we actually already have all of these numbers over here, and we've calculated the different p ones and p twos in a previous video, so we could just go ahead and use those numbers. If not, you would just have to calculate them really quickly by just doing x over n for both of the samples.

So basically, what we're going to do here is we're going to do p1^ - p2^, which is 0.55 - 0.74, which gives you a point estimate of -0.19.

In other words, there is a 19% difference in these two samples, and this is sort of your best initial guess right now.

Now that we're done with the point estimator, we're going to go ahead and find the margin of error. So the margin of error equation is e, and it's just going to be slightly different for a two sample proportion. Basically, it's still going to be a critical z score times something in a parenthesis, but now we're going to have to use both of those samples.

Now what goes inside of here is going to be p1^ and q1^, and then in the second term, you're going to have p2^, q2^. So in other words, it's always the ones with the ones and the twos with the twos.

You're just going to go ahead and multiply those things, add them up, and then take the square root and multiply by the critical z score.

Now one thing to note here is that unlike what we used in the hypothesis test, we're going to use the individual sample proportions. In other words, the p1^ and p2^ instead of the pooled sample proportion. Remember, that was p bar. We don't use that in the margin of error equation.

We don't need that p bar. So we're going to move on to step four, which is calculating the margin of error. So e equals the critical z score, z = 1.645 times, and then we're going to have this big square root over here. Now this is going to be 0.55 times the complement, which is 0.45.

Remember, if you already have these p ones, you can calculate the q ones, the q one hats. This is just going to be 0.45. And then for q two hat, this is just going to be, this is going to be 0.26. So this is going to be 0.45 * 0.55 divided by the n one, which is going to be your 20.

Right? This is your n one, and this is your x one. And then this is going to be, x n two and x two. So this is going to be plus, and then you're just you're going to do all of the second sample data.

Right? This is just going to be 0.74 * 0.26 divided by n2, which is the 23. So all of this stuff goes into your margin of error equation and you should find a margin of error of 0.24.

So now that we have our margin of error and we've calculated our point estimate, we just put all of that stuff together exactly how we would in a one sample confidence interval to basically build our upper and lower bounds.

You're just going to take your point estimator from step three and subtract the margin of error and then add it to form your upper your lower and upper bounds respectively. So this is basically just going to be our, our parenthesis is going to be -0.19 - 0.24. That's going to be the lower one.

And on the upper one, we're going to have -0.19 + 0.24. That's our upper bounds. Now if you go ahead and work all of this out and plug those numbers in, what you should find is that your confidence interval is -0.43 to 0.05. This is your final answer for your 90% confidence interval.

So this is going to be 90% CI. Okay? So this is your correct answer right over here. I promise we would get back to the second part of this problem over here, which is, now that we have this confidence interval, what does the result suggest about the claim that there is no difference in proportion between the two groups?

So basically, a lot of times in these problems, you may have to use a confidence interval and connect it to a claim that's being made, in other words, a hypothesis test. And here's the connection. Basically, what's going to happen here is that in hypothesis tests, your default assumption was that the difference in proportion was always zero. So the number you're always going to look for in your confidence intervals is just going to be the number zero.

And here's the rule for this. If your confidence interval doesn't include zero, then that means that you're confident of a difference between those two proportions. So if your confidence interval does not include zero, what you're going to do is you're going to reject the null hypothesis. There's evidence of a difference between those numbers if you don't have zero. Now if your confidence interval does include zero, then it's actually possible that there is no difference between p one and p two.

Therefore, what you're going to do is you are going to fail to reject the null hypothesis. So it's a little bit of a double negative here, but if it doesn't include, you always reject, and if it does, you always fail to reject. So let's go back to our interval over here. So we have a -0.43 and a 0.05. Zero goes somewhere inside of that interval over here, so this does include zero.

So, again, you may not have to write these big fancy sort of paragraphs, but in your course if there are some stricter requirements, I would go ahead and write it. Basically, here's how this would go. So we are 90% confident here that the true difference in proportions of people who quit using these two different approaches is between -0.43 and 0.05. Now because this interval does include zero, then what happens is that we fail to reject the null hypothesis that these two are actually equal to each other. Basically, what happens is there's not enough evidence that there is a difference in those two proportions and yada yada yada.

You would explain the rest of it in sort of paragraph form. Alright? So, that's really all you have. Let me know if you have any questions, and thanks for watching.

A researcher using a survey constructs a 90% confidence interval for a difference in two proportions. According to the data, they calculate $$${\hat{p}}_1-{\hat{p}}_2=0.09$ with a margin of error of 0.07. Should they reject or fail to reject the claim that there is no difference in these two proportions?

• A. Reject
• B. Fail to reject
• C. There is not enough information to answer the question.

Answer: (A)

The data below is taken from two random, independent samples. Calculate the margin of error for a 99% confidence interval for the difference in population proportions.


$x_1=87$, $x_2=68$
$n_1=120$, $n_2=115$

• A. 0.061
• B. 0.062
• C. 0.158
• D. 0.060

Answer: (C)

Difference in Proportions: Confidence Intervals Example 2

Welcome back, everyone. So let's go ahead and take a look at another problem where we have to construct a confidence interval from scratch, from start to finish, using a problem where we're given two proportions. So we're going to go through all of the steps. It's going to take us a little bit of time, but just bear with me, and we'll get through it together. Let's get started.

We have a university that wants to know if students who live on campus are more likely to attend campus events than students who live off campus. So they take two random samples for this. The first one is a sample of 50 on-campus students, where 102 attend at least one event. And then in the second group, there are 30 off-campus students, of which 74 attend at least one event. Right?

We're going to construct a 95% confidence interval. So that's the first part of this problem over here for the difference in proportion. And then we're going to use that in the second part of the problem to test a claim that's being made about these proportions. Alright? So let's go ahead and get started here.

We're going to stick to the steps. The first thing we have to do is just check some really quick conditions. We just have to verify for each sample that these samples are random and independent, which almost always you can assume that they are unless you have some glaring obvious reasons why they aren't. In this case, there's a random sample of a bunch of students, and they don't affect each other, so they're independent. We also just have to check if there's more than five successes and failures.

And if you look at these numbers, 50 and 102, 30 and 74, it's very clear that we can assume that these are normally distributed. Alright? So in other words, we're going to go ahead and take a look at the second step now. Remember, we have to figure out what that critical z value is. This works exactly the same for a confidence interval in any other type of problem here.

So let's get started here with the second step. We're going to find this critical z value, which remember comes directly just from our confidence level. It has nothing to do with a problem with a number that's given to you, like any one of these numbers over here about the samples. Alright? So the confidence level we were told here is we're supposed to do a 95% confidence interval.

So in other words, our confidence level is 0.95. This gives us a z score or a critical z value of you take this, subtract it from one, and then divide it by two. So you're basically looking for what z score will give you two and a half percent on either side of the normal distribution. This is a number that we've seen before, should look pretty familiar, but it's 1.96. Right?

So that's the second step. We've already verified some conditions and found a critical z-score. Now remember, in order to construct a confidence interval, first we have to do is figure out what's the first best guess, what's the middle of our confidence interval. That's going to be your point estimator, which is just going to be basically pₕ hat minus pₖ hat. This is where we use a lot of the sample data that we're given in our problem.

So let's go ahead and move on to the third step over here, which is in order to find out what our pₕ hat minus pₖ hat are, let's just go ahead and get all of our information organized. I've got these two groups of students, and I'm just going to sort of label them. These are going to be my on-campus students. I'm going to call that group one, and then these are going to be my off-campus students. This is going to be group two.

Alright? So I can see here that I'm just going to call one xₕ nₕ and xₖ nₖ. So my xₕ, is just going to be the number of successes in the first group. I'm going to label success as attending at least one campus event. So in other words, this is going to be 102, and the total sample size or number of trials in the first group is just going to be 50 because there's 50 on-campus students.

Alright? Now for the second one, the number of successes, this would be xₖ, is going to be 74, and then nₖ is going to be 30. Alright? Now remember, we're going to need to calculate pₕ hat and pₖ hat. We'll also need it when we get to the margin of error calculation.

So it's always a good idea to calculate these things as soon as you have these numbers. You're just going to do x over n, so 102 divided by 50, which would give you 0.68. Just going to keep it to three decimal places. Alright? Same thing for xₖ over nₖ.

So in other words, this is going to be pₖ hats, 74 out of 30. Let me go ahead and work this out. This is going to be 0.569. Alright? Now remember, ultimately, we want to get to what is basically pₕ hat minus pₖ hat.

That's our point estimator. You can basically just subtract these things. But one other thing you might want to do as you're doing all these calculations and you have your calculator handy is you want to calculate qₕ hats because we'll need them for the margin of error equation anyway. So it's always just a good idea to get all these calculations out of the way first. Basically, this is just the complement of 0.68.

You could just subtract this from one, and you're going to get 0.32. And then over here for qₖ hat, you're going to get the complement of this, which is 0.431. Alright? Now that you're done here, you can go ahead and calculate what the pₕ hat minus pₖ hat is. This is just going to be the 0.68 that we just found minus 0.569, and you're going to get a point estimator of 0.111.

This is basically the middle of your confidence interval. It's your best guess for what the difference in proportion is between these two groups. Alright? So that's the third step over here: calculating the point estimate. Now remember, for a confidence interval, you're going to have a point estimate, the middle of your interval, and then you need to subtract and then add the margin of error in order to build your bounds, which is basically the fourth step over here.

We're going to calculate our margin of error using that new equation that we just learned in a previous video. Alright? Now if you don't have it handy, I'm just going to go ahead and write it out real quick, but you should have it handy. You won't be expected to memorize it, but it's basically just going to be zα/2, sorry, all critical z score, times in parentheses our square roots, we're going to have pₕ hats qₕ hats over nₕ plus pₖ hats qₖ hats over nₖ. So in other words, the ones are the ones, the twos are the twos, so on and so forth.

Right? So if you go ahead and work this out, what you're going to get here is this is going to be 1.96, our critical z value, times, and then this is going to go in parentheses, and you're just going to plug in all of these numbers that we just found right over here. Okay? So your pₕ hats is going to be 0.68, times qₕ hat, which is going to be 0.32. This goes over nₕ, which was 50.

And then and just make sure you have a lot of room when you're doing this. You're just going to do plus, and this is going to be 0.569 times 0.431. Just plug in all the numbers that we just found over here, divided by the 30. Alright? Just go ahead and plug all that stuff carefully into your calculator.

But, basically, what you should find for the margin of error is just going to be 0.113. Alright? Now just weirdly, sort of coincidentally enough, this happens to be pretty close to our point estimate, but that's, again, purely by coincidence. It's not always going to happen, but that's your margin of error: 0.113. Alright?

Right. So now that we're done with our margin of error, now we have our point estimate and a margin of error, we just put them together in the last step to basically figure out where your upper and lower bounds are. You're just going to take your pₕ minus pₖ hat, subtract e, and then add e, and that's going to form your interval. Okay? That's what I'm going to do here in this very last step over here, which is going to be, step five.

Alright? So, basically, what you're going to do here now is you're going to take your 0.111 minus 0.113. That's going to be your lower bound. And then 0.111 plus 0.113. That's going to be your upper and lower bounds.

And, basically, what you should find over here for your 95% confidence interval, this is going to be 0.0224. Alright? So we got 0.0224. That's going to be our confidence interval. Alright?

So that's the first part of the problem here. That's how to construct the 95% confidence interval. Alright? So this is just going to be a 95% confidence interval. That's the first part of the problem.

Now let's take a look at the second part of the problem here, which is, again, how we tie this back to a hypothesis test or a claim that's being made about these differences in proportions. So using this confidence interval that we just found here, can we test the claim that the on-campus students are more likely to attend events than off-campus students? In other words, is there actually a difference in these two proportions? That's basically what this claim is being said. Now remember, when it comes to these types of problems, it always just depends on whether zero is in this interval or it's not.

The claim is that basically there is no difference, that the null hypothesis is that there is no difference in the proportion between students who attend these events, whether they're off-campus or on-campus. So what happens here is you just take a look here in this sixth step. I'm just going to write it out over here. You can choose to write out the whole entire thing, but basically what happens here is, we can see here that this CI or this confidence interval, does include zero. So because it includes zero here, that means that it's plausible that there actually is no difference in these proportions, and we're 95% confident that the, actually, the real difference in proportion could, in fact, actually just end up being zero or no difference.

So in other words, the CI does include zero. Therefore, what happens is we fail to reject. So remember when we when we when the interval does include zero, we always fail to reject because it means it's possible that there actually isn't any difference in these proportions, which is the null hypothesis. So we fail to reject it. Therefore, there is not enough evidence.

That's what you would say about the alternative hypothesis, which is the claim that's being made is that on-campus students are more likely to attend than off-campus students. So there's not enough evidence to support that. Alright? So, again, there's a more complete way to write out all of that statements. You should always just double-check with your course and see if you're expected to do that, but I'm just going to shortcut a little bit.

But you would fail to reject, and there wouldn't be enough evidence. Alright? So that's it for this one, folks. A lot of different steps, but just stick to the steps, and you'll get the right answer every time. Let me know if you have any questions.

Thanks for watching.

Here’s what students ask on this topic:

What is the null hypothesis in a two-proportion hypothesis test?

In a two-proportion hypothesis test, the null hypothesis states that the two population proportions are equal. This is mathematically represented as $H_0:p_1=p_2$. The null hypothesis assumes that there is no difference between the two proportions being compared. This serves as the default assumption, and the goal of the hypothesis test is to determine if there is enough statistical evidence to reject this assumption in favor of the alternative hypothesis, which suggests a difference between the proportions. Understanding the null hypothesis is crucial as it forms the basis for calculating the test statistic and making conclusions about the data.

How do you calculate the test statistic for a two-proportion hypothesis test?

The test statistic for a two-proportion hypothesis test is calculated using the formula: $z=\frac{(p_1-p_2)}{\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}}$. Here, $p_1$ and $p_2$ are the sample proportions, and $n_1$ and $n_2$ are the sample sizes. This formula helps determine how far the observed difference in sample proportions is from the hypothesized difference (usually zero), in terms of standard deviations. The resulting z-score is then used to find the p-value, which indicates the probability of observing such a difference if the null hypothesis is true.

What is a confidence interval for the difference in proportions?

A confidence interval for the difference in proportions provides a range of values within which the true difference between two population proportions is likely to fall. It is constructed using a point estimate (the observed difference in sample proportions) and a margin of error. The formula for the confidence interval is: $(p_1-p_2)\pm z\surd \frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}$. The z-score corresponds to the desired confidence level (e.g., 1.96 for 95% confidence). If the interval includes zero, it suggests that there may be no significant difference between the proportions. Conversely, if zero is not included, it indicates a significant difference.

How do you interpret a confidence interval that includes zero in a two-proportion test?

In a two-proportion test, a confidence interval that includes zero suggests that there is no statistically significant difference between the two population proportions. This is because zero represents the point where the two proportions are equal. If the interval spans zero, it indicates that the true difference could be zero, meaning the observed difference in sample proportions could be due to random sampling variability rather than a true effect. Consequently, we fail to reject the null hypothesis, which states that the proportions are equal. This interpretation is crucial for making informed decisions based on statistical evidence, as it helps determine whether observed differences are meaningful or likely due to chance.

What are the conditions for conducting a two-proportion hypothesis test?

To conduct a two-proportion hypothesis test, certain conditions must be met to ensure the validity of the test results. First, the samples should be randomly selected and independent of each other. This means that the selection of one sample should not influence the other. Second, each sample should have at least five successes and five failures to approximate a normal distribution, which is necessary for the z-test. These conditions help ensure that the sampling distribution of the difference in sample proportions is approximately normal, allowing for accurate calculation of the test statistic and p-value. Meeting these conditions is essential for drawing reliable conclusions from the hypothesis test.