Two Means - Matched Pairs (Dependent Samples): Videos & Practice Problems

```html

Matched pairs are samples that are related in a unique way, often involving a before-and-after comparison of the same individual. In hypothesis testing, matched pairs simplify to a one-sample mean test by using the mean difference ($d_{̄}$) and standard deviation of differences ($s_d$). Confidence intervals for matched pairs focus on the population difference (${\mu }_d$), using the sample mean difference as a point estimator. This approach helps determine the effectiveness of interventions, such as medication, by analyzing changes in paired data.

```

Introduction to Matched Pairs

One of the things we always check for when doing hypothesis tests with two samples is that these samples were random and independent, meaning they didn't interact with each other or affect each other in any way. But what if those samples actually did affect each other? Well, that's what we're going to talk about in this video because there's a term for that. Those samples are called matched pairs. Now, eventually, I'm going to show you how to do a hypothesis test with matched pairs.

But in this first video, I'm just going to show you how to identify whether two samples are matched pairs in the first place. So there are a couple of things to watch out for. We're going to jump right into our example and we'll do a bunch more practice problems. Let's get started. So, we're going to start with the basic definition here.

What does it mean for two samples to be matched pairs? Well, basically, those two samples are matched pairs if they are related to each other in some unique way. And we're going to talk about that in just a second. Moreover, each value from one sample has to be paired to another in the second sample in a one-to-one relationship. That's why we call them matched pairs.

You need to be able to pair up two values from both of the samples. So, let's go back to what it means for two samples to be related to each other. Some very common relationships you want to look out for are when you do a before and after comparison of the same object or individual. For example, you take a test one week and then take the same test the second week to compare your test scores. That's a relationship between before and after scores, and it's the same individual.

This is the most common type of relationship that you're going to see. Now, the second one, a little less common, is where you actually have literally related individuals in some connection. They could be either siblings, coworkers, partners, things like that. The last one, even less common, is where you have data that is self-reported for some object or person versus measured data of some object or person.

So, again, the most common one you're going to see is this first one over here. So, let's go ahead and take a look at our example here. Just some pretty basic criteria to check for if some things are matched pairs. All right? So, we have these two examples over here.

We're going to determine if these samples are matched pairs. And if so, we're going to calculate some differences and things like that. But we're going to get to that in just a second. Alright? So, in part a, we have data below that shows the heart rates of a sample of nine adults before and after sleeping.

So, we have these nine samples or these nine numbers corresponding to their heart rates in beats per minute before and then after sleeping. Alright? So there are a couple of things to watch out for when you're determining something is a matched pair. The first one, the easiest thing to check for is that the two sample sizes have to be the same. Remember, each value from one sample has to be paired with another in a one-to-one relationship.

So if you have nine numbers over here and eight numbers, they can't be paired up one-to-one. Alright. So, clearly we can see here that the sample sizes are the same, and generally, this is usually going to be the case. However, just because two things are the same sample size doesn't mean that they're matched pairs. There are a few other things to watch out for.

The second thing you want to look out for is that the samples are related to each other according to one of these scenarios that we just covered up here, one of the three over here. Alright. There are a couple more, but these are generally the most common ones that you're going to see. So are these samples related to each other? Well, we're comparing heart rates in beats per minute between the same exact individuals, just in a before and after relationship.

So the heart rate of individual one is 84, and then afterwards, it's 80. We're comparing before and after of the same person. So these samples are definitely related to each other in some way. Alright. Now the last thing, and this is usually the hardest one to sort of watch out for or sort of identify, is that the values have to be paired up in a one-to-one relationship.

What that means here is that I have to be able to physically link these two values together. So, for example, does it make sense to compare the before heart rate of individual one to the after heart rate of individual three? It doesn't make sense to do that. We're comparing the heart rates of an individual before and after sleeping. So, there is a literal connection, a one-to-one relationship between each one of these values.

Alright. So, if these all these three criteria apply, then this is definitely going to be a matched pair. Alright. So these are definitely matched pairs. Let's take a look at the second one because the setup is very similar, but there are a few important differences.

In part B, we have the data below. It's again heart rates in beats per minute, the exact same thing. But, we have a sample of adult males and females. We have nine males and nine females. Alright.

So, we've got eight heart rates in beats per minute. Presumably, these are like averages or something like that. And then I've got nine for females. So again, let's go through the checklist. Are the two samples the same size?

They are. We have nine in both values in both samples. So these are definitely the same samples or the same sample sizes. However, that doesn't automatically mean they're matched pairs. So, are these samples related somehow?

So, we have here that we're basically comparing heart rates, the same exact thing, beats per minute, between a group of nine males and a group of nine females. So it's not a before and after relationship because we actually have two different relations. You have two different individuals, so it's definitely not relationship one. Are these things related to individuals somehow? Is it like siblings or coworkers or partners or something like that?

It's none of those things. They're not related to each other. It's just a group of nine random adults and males and females. And this is not self-reported or measured data. So even though we're comparing the same exact thing in B and A beats per minute, these samples aren't related somehow in any of these unique ways that we've seen.

Alright. The last thing again is you want to check if the values are paired in a one-to-one relationship. So if you take a look, what happens here is can you actually compare the heart rate of male one to the heart rate of female four? You absolutely can. There's nothing that's really preventing you from doing that.

So there's really nothing that suggests these values are paired in a one-to-one relationship. Alright? So therefore, because of these two conditions that fail, this is definitely not a matched pair. These are actually just independent samples. And again, it's usually going to happen when you have different sorts of individuals that are going on between the two samples.

All right. So this is not a matched pair on the right. So now let's take a look at the second part of the problem over here, which is if these things are matched pairs, we're going to calculate a couple of things. Now one of the things that we calculate commonly with matched pairs is the difference in those values. The letter that we use for this is just the letter small d or little d.

And this is just going to be where you just take one value and subtract another one between each matched pair. All right. So let's go ahead and do this here. Now, whenever you do this, you should always just make sure that you're subtracting in the same order. Most of the problems will define which order you're supposed to do it in, like a before, minus, and after or something like that.

Alright. It's usually pretty obvious in the way that they're doing those subtractions. So in this case, what we're going to do over here is we're going to do 84 minus 80, which is four. And once you do this, you should always just do it in the exact same order for all of the pairs. So 70 minus 73 is negative three.

Perfectly fine to have differences that are negative numbers or even possibly zero. So, this is going to be negative 10. We've got negative 12, then we've got positive two, then we've got eight, then negative one, negative 16, and then negative 23, always subtracting in the same order every time. Alright? So, now after we've calculated the difference for each one of these matched pairs, which you may have to do in problems, you're going to have to find the mean difference.

Now the mean difference is given by the letter d bar. It's very similar to how we used X bar for the sample mean when we only had one sample. But now that we have two, we calculate the difference and so we just use d bar. Okay. So when you go over here and you calculate this d bar, now there are a couple of different ways you could do this.

You could just do this all by hand. Works the exact same way. You're going to add up all of these numbers and divide by the total number that there are. However, you can also just use technology to make your life a whole lot easier. We've already covered how to calculate means and standard deviations.

It's not what we're going to cover in this video. So, I'm just going to go ahead and show you the five-number summary for plugging all of these numbers into a calculator. What we can see over here is that if I have my calculator, I've got my sample, I've got my mean, which is negative 5.667. I'm just gonna go ahead and put this as negative 5.67. Then we have a standard deviation.

The letter we use for this is SD. That little d subscript indicates that we're calculating the standard deviation of the distances of the difference. And we're going to use this sx that's given to us in this readout because, remember, it usually gives you a population and a sample standard deviation. So we're just going to read off this number over here, which is going to be 10.21. All right.

So this is the sample mean difference and the standard deviation of this data set. All right. All right. So that's really all there is to it. Let me know if you have any questions.

Let's get a bunch more practice with figuring out if something is a matched pair or not. Thanks for watching.

Introduction to Matched Pairs Example 1

Figuring out whether two samples that you're given are independent or matched pairs is one of the most important skills in this topic because it's going to affect the type of hypothesis test you're going to perform. So as promised, let's take a look at a couple more examples and see if we can determine which one of these is independent and which one are matched pairs. Alright? Remember, we have a couple of things that we can check to help guide us in the right direction, but let's just take it from the top.

So, in this first problem, we have a nutritionist who measures the blood pressure of 30 different patients, 30 patients before and after they begin a new diet with the goal of determining whether the diet has an effect on blood pressure. Remember, there are a couple of things that we're going to look for here. Are the sample sizes going to be the same? Well, what’s going to happen here is the nutritionist is going to measure the blood pressure, 30 patients before and after.

So, in other words, there are going to be 30 data values before, and then there will be 30 data values afterward. This means the sample sizes of the before and after are going to be the same. That's definitely one thing. If they're not the same, that's a dead giveaway.

This is not going to be matched pairs. But, just because they are equal, we're still going to keep going and seeing if the other criteria fit. Now, are the samples related somehow? This is the trickiest one to figure out because there are non-exhaustive ways things can be related to each other. But the most important ones that are most common are before and afters, any type of relationship, like personal relationships like siblings or partners or friends, coworkers, or something like that.

The other one that's really big is whether data is self-reported versus measured. Essentially, you have the same type of data measured under a few different circumstances. Clearly, we can see here that there is a before and after relationship, which is almost always a dead giveaway that this is going to be a matched pairs. We're also looking at the same exact patients, the same individuals.

We're just measuring before and after. Now, are the values paired one to one? Basically, what we're doing here is determining whether the diet that they're on in the experiment has an effect on their blood pressure. In other words, you're going to pair one of the values from the before sets to one of the values from the after and seeing if it has an effect.

Clearly, these values are paired between the sets. There is a one-to-one relationship between each value in the before and each value in the after. These are important things to sort through as you're determining which ones are matched pairs. So clearly, we can see here that these are matched pairs.

Let's take a look at the second problem. So, in the second problem here, a teacher gives 40 students a practice exam, and then a week later gives a different group of students, forty students, the same practice exam.

The teacher wants to know the difference in the scores. Clearly, we can see here that there are two groups. We have \( n_1 \), the sample size is 40, and we have \( n_2 \), the sample size is also 40. So let’s take a look at our checklist here. Does \( n_1 \) equal \( n_2 \)? Clearly, yes, they are. So, is there a type of before and after relationship or a type of sibling relationship?

Not exactly siblings, but they definitely are two groups of people doing the same thing, which is taking a practice exam. I'm going to check that the samples are related because the different students are taking the same thing, the practice exam. Now, let's look at the last one here, which is whether the values are paired one to one. This is where you have to assess whether each value in set one is paired with another value from set two.

This is where this starts to fall apart a bit. The teacher wants to know the difference in the scores. Basically, we're going to look at all of the 40 students who take the exam, and you're going to compare it to a different group of 40 students taking the same exam. What’s happening here is that there’s no relationship between each individual pair as there was in the before and after. I could take each one of these data points and link it to another one in the after set.

Whereas, I can't take a data point from one student one week and link it to one student the following week because they're two different students. So, these values are not paired in a one-to-one relationship. Therefore, this is not matched pairs. This is actually just two independent samples.

So, this is going to be independent. Let’s move on to the third question. This is going to be really similar to the second one, but it's a little bit different. A teacher gives 40 students a practice exam, then a week later gives the same 40 students a similar graded exam, and the teacher wants to know if the practice improves scores. So, again, is \( n_1 \) equal to \( n_2 \)?

We basically have \( n_1 \) is 40, and then a week later, the same 40 students. In other words, that’s \( n_2 = 40 \), so the two sample sizes are the same. Now, are the samples related to each other? Just like before, they're both taking that type of exam from week to week, so it definitely looks like there is a relationship. And now the big one: are the values paired one to one?

Clearly, we can see here that 40 students take an exam one week, and then the same exact 40 students take it again the next week, but now it's under different circumstances. What happens here is that the teacher wants to know if practice improves scores for each individual student. In other words, there is a link between each one of these scores in the before case to an individual score in the after case for each individual student. Therefore, there is a value of a one-to-one pairing of the values. This is definitely going to be a matched pair.

So this is a matched pair over here, and over here, but the second one was not. The second one failed. These are independent samples.

So, hopefully, this makes a little bit more sense in figuring out what types of scenarios are independent and which types of samples are matched pairs. Thanks for watching. Let's move on.

A personal trainer is studying whether a new stretching routine improves flexibility. She records the forward reach (in cm) of 6 clients before and after a 4-week program. Calculate the difference (after - before) for each client, the mean difference, and standard deviation.

• A. Difference for each client: $A=4,B=4,C=3,D=-1,E=0,F=2$; The Mean Difference = $2$; and Standard Deviation = $1.91$
• B. Difference for each client: $A=4,B=4,C=3,D=-1,E=0,F=2$; The Mean Difference = $2.5$; and Standard Deviation = $1.91$
• C. Difference for each client: $A=4,B=4,C=3,D=-1,E=0,F=2$; The Mean Difference = $2.5$; and Standard Deviation = $2.10$
• D. Difference for each client: $A=4,B=4,C=3,D=-1,E=0,F=2$; The Mean Difference = $2$; and Standard Deviation = $2.10$

Answer: (D)

Matched Pairs: Hypothesis Tests

Now when I first introduced matched pairs, I said we would eventually use them in hypothesis testing. Well, that's exactly what we're going to cover in this video. Now you might be thinking that this is going to be really complicated because we have two different means, and there's going to be changes between them like a before and after, but luckily none of that's true. Because, really, all there is to it is that this works exactly the same way as it did for a one-sample mean test. Alright?

So let's just jump right into this problem here, and I'll show you how this works. Now the basic difference between a matched pairs problem and any other type of means test is that when we use matched pairs, we're basically just going to replace all of our normal one-sample variables like x̄, μ, and s with their difference equivalents, with their d equivalents. So x̄ just becomes d̄. We're going to replace μ with μ_d. That's the mean difference.

And then s with s_d. That's the difference the sample standard deviation of the difference. Alright? Other than that, all of the same steps are exactly the same. So let's just jump right into this problem here, and we'll do a whole hypothesis test.

Now a company is claiming that its new medication reduces blood pressure. So in a study, they're going to take the sample mean difference in 37 different patients' blood pressure before and then fifteen days after this medication. Now what they find here is that the sample mean difference is 3.2 with a standard deviation of 6.7. Alright? So we're going to perform a hypothesis test with alpha equals 0.05 to determine if the medication is effective in reducing blood pressure.

Alright? So we basically have some kind of a before and after. This is definitely a matched pair. Let's go ahead and take a look at the steps. Now we have some basic conditions to cover here.

Are these random samples, and are they matched pairs? Well, they're definitely random samples, and we know this is a before and after using the same number of people, so it's definitely a matched pairs. Right? We're trying to see if there is a one-to-one sort of mapping in reducing blood pressure. Alright?

And last thing we just check if this is if the samples are normally or population is normally distributed or we have a big enough sample size. In our case, we have 37, so that condition is definitely met. Alright? Now let's take a look at the first step over here, which is writing our initial hypotheses. Now when we did this for one mean, remember, we always just sort of picked a value that we found from the problem, and our default assumption was that the mean was equal to that value.

So how does it work now when we have matched pairs, especially in a before and after type situation? Well, what happens here is we're looking to find evidence for if the medication is effective in reducing blood pressure. So what would the default assumption be? The default assumption is that there is no difference in change in blood pressure. Alright?

So, in other words, how would you actually write that as an initial hypothesis? Remember, this is going to be the sample mean difference, that's μ_d, and we're just going to say the default assumption is that it's nothing. Right? There is no change, but there is no difference. Alright?

So that's, sort of by the way, this is usually going to happen with a before and after type of matched pairs test. Alright? So what is your alternative hypothesis? Well, we're still going to have that and we're going to have to compare it to zero, but now there's a sign that goes in here. Is it a less than, greater than, or not equal to?

Now you might be looking at this term over here, which is reducing blood pressure and think that it's going to be a less than sign. However, this is what's important to keep track of in a before and after type situation. So we're told here that the sample mean difference is 3.2, and that's between the blood pressure before and then fifteen days after. So in other words, what happens is the blood pressure was higher before and then lower afterward, meaning the difference is a positive number. That's what we're actually looking to find evidence for.

So even though this says reducing blood pressure and you think that's going to be a less than sign, because the difference is calculated as a before minus an after, then a positive number actually is reduction in blood pressure. So you're actually looking for evidence that μ_d is greater than zero. That's your alternate. Alright? A little bit tricky here that won't always happen, but it's always good to reason through.

Alright? Now that's the first step. The second thing we're going to do is go ahead and calculate our test statistic. And, really, it's going to be very similar for a one-sample test versus a two. We just replace a couple of variables.

Alright? Now let's just go ahead and fill out our information. I like to organize everything in a little column over here. So what's our n, our sample size? Well, in this case, what happens is because we're looking at matched pairs, really, the n that this refers to in our equations is just the number of pairs that we have.

We have 37 patients who will have before and after data, which means 74 data points, but it's just 37 pairs. Alright? So they're not going to try to trip you up that way. It's we're usually going to be pretty straightforward. Alright?

So what about this d̄, the sample mean difference? We're actually told explicitly what that is. The before and after was a sample mean difference of 3.2, and the standard deviation for the difference was 6.7. Alright? All that stuff is pretty straightforward.

So now we're just going to plug this straight into our t-score equation. We use this again because we don't know what the population standard deviation is. Okay? Now remember how this goes? Basically, you're going to replace x̄ with d̄.

So, in other words, the first thing that goes here is what's the d̄? It's just the 3.2. Alright. Now we're going to subtract. And the second thing, remember, we always had μ, and this μ came from whatever your null hypothesis was.

It works the same exact way in matched pairs. So what's the μ_d that we set over here? We set it equal to zero. So, really, what goes inside this parenthesis is just zero. We're assuming here the null hypothesis is that μ_d is equal to zero in this case, so we just set that equal to zero.

Alright. So now we're going to subtract, and I'm going to divide by the standard deviation. This is just going to be s_d over n. So this is just going to be our sample our different standard deviation, which is 6.7 divided by the square root of n, where n is 37, the number of pairs. Alright?

If you go ahead and work this out, what you're going to find is a t-score of this is going to be 2.91. Alright? So 2.91, and, again, the rest of this works exactly the same way as any other hypothesis test. Once we find the t-score, we're going to go ahead and convert this into a p-value. Alright?

Now in order to do this with a t-distribution, first, we need to figure out what's the degrees of freedom that we have. The degrees of freedom works exactly the same way as it does any any other type of problem. You're just going to take the n, subtract it by one, so our degrees of freedom is 36. This is approximately going to be like a normal distribution, and our p-value, because we used a μ_d is greater than zero, this ends up being a right-tailed test. So our null or sorry, our alternative hypothesis is going to tell us what type of test to do.

We're looking at the distribution on the right side. So basically, what we're looking for here is what is the probability that t is greater than the number that we calculated over here, 2.91. Alright? So what you go ahead and find here, again, you can use a calculator or an online resource, however you want to do this, you should find here is a probability of 0.0031. That's your p-value.

Now remember, the last thing that we do here in any hypothesis test is compare it to the alpha, which is your level of significance. This is always given to you in the problem, and in this case, it was 0.05. So our alpha was equal to 0.05. Clearly, we can see here that our p-value was much, much smaller than that. So this sort of brings us to our conclusion.

So, again, works the same exact way as any other hypothesis test. Because our p-value was less than our alpha, we reject the null hypothesis, which, in other words, means that there is enough evidence to suggest that the medication is effective in reducing blood pressure. Alright? So, that's really all there is to it. So here we have a complete rundown of a hypothesis test using matched pairs.

Let's go ahead and take a look at some practice.

Matched Pairs: Hypothesis Tests Example 2

So let's get some more familiarity with a hypothesis test using matched pairs. Alright. Now, in this problem, we're given some information and we're not going to do the entire hypothesis test from start to finish. We're just going to focus on the first two steps. Let's check it out.

A professor is going to randomly select 12 students. So our sample over here is going to be random and records the scores on a pop quiz versus a scheduled quiz. So they have two quizzes given in the same week. Now, each student's two scores are going to be paired together, and the difference is calculated as however they scored on the pop quiz versus however they did on the scheduled quiz later on. Now, from the data, this professor calculates that the mean difference, remember that's \( \bar{d} \), is equal to negative 1.1 with a standard deviation of 1.8.

Alright? So in other words, we have that \( n \) is equal to 12. Remember, we have 12 pairs of data. Each is going to be a matched pair. We compare the difference in their pop quiz versus their scheduled quiz. So, \( n = 12 \), \( \bar{d} = -1.1 \), and the standard deviation of the difference is 1.8.

Now, from this data, we calculated these two things, and we're going to write the null and alternative hypotheses at step one of any hypothesis test. And then we're going to calculate the test statistic. Alright. Let's go ahead and take a look at the first step over here. You're always going to write your \( H_0 \) and \( H_a \).

Now, how does this work for a hypothesis test for means or, for matched pairs? Well, remember, ultimately what we're using is we're using a small sample of 12 students to figure out basically how the whole population of students is going to perform on this pop quiz versus this scheduled quiz. Alright. So we always use the letter \( \mu \) for means. And the only thing that's different here when we're looking at matched pairs is we just use the letter \( \mu_d \).

Again, usually what happens in these types of problems is your default assumption, the status quo, is that there is no difference in the difference between the two scores. Alright. So, \( \mu_d = 0 \) is going to be your default assumption. Now, for your alternative hypothesis, you're just going to figure out what's the sign here? Is it a less than, a greater than, or a not equal to?

If you take a look at this problem here, we're going to write our hypothesis to test a claim that students perform worse on the pop quizzes than scheduled quizzes. So this is the language that's going to tell you which sign to put here between \( \mu_d \) and zero. Alright. Now, worse on pop quizzes, if you take a look at how we calculated the difference, the difference was calculated as the pop quiz minus the scheduled quiz.

So, if they perform worse, that means that this score over here, the first one, is less than this one, and therefore you're going to end up with negative numbers. Alright. So basically, the claim that's being made here, the alternative hypothesis, is that \( \mu_d < 0 \). This is just going to be a left-tailed test.

This would be useful if you were to find a p-value and go on with the rest of the conclusion. But we're not going to do all that. Alright. So those are the two statements that we have for \( H_0 \) and \( H_a \). Now let's take a look at the test statistic. Again, this works very similarly to how we did this for a one-sample means test.

We're just going to replace a couple of variables. Alright? Now, if you don't have the equation handy, remember, instead of \( \bar{x} \) like we did for a one-sample mean, we're just going to use \( \bar{d} \) for the mean difference. Right? Now, instead of \( \mu \) for the one-sample population mean, we would just use \( \mu_d \) for the population difference in the mean.

And for the formula, on the bottom here, we have the standard deviation of the difference divided by the square root of \( n \). Alright. So again, very similar to how this works. We just replace a couple of the letters with their difference equivalents.

So we plug and chug all the numbers that we have. So we have \( \bar{d} \), which is the difference in the small sample that we had, which is -1.1. That's already given to us in the problem.

And then for \( \mu_d \)? Remember, this value in our test statistic always will come from whatever our null hypothesis is. Most of the time it's going to be zero. So, it's just going to be zero there. Alright. So then over here, for the standard deviation, we have 1.8 divided by the square root of 12.

Plug this into your calculator, and what you should find for the test statistic is that this is equal to -2.12. And that's the answer. So we have our null hypothesis, we have a test statistic. And again, we could continue on and calculate a p-value for this as long as we knew the degrees of freedom, and we could go on and write a conclusion here.

We don't have to do any of that. One final thing here, by the way, is that whenever you calculate the test statistic for \( t \), you don't actually need the degrees of freedom for that. That's something you use in the next step. Alright. So that's it for this one, folks.

Let me know if you have any questions. Alright.

Matched Pairs: Confidence Intervals

So earlier when we learned matched pairs, we learned how to identify whether two samples were matched pairs or not, and then we used them in hypothesis tests. And everything worked exactly the same as it did for a one-sample test; we just replaced a couple of variables. Similarly to other types of tests we've seen before, some questions will ask you to create a confidence interval for a mean difference using matched pairs. But luckily, we already know all the steps to creating a confidence interval, and nothing's really different when it comes to matched pairs. We'll just need some slightly different equations when it comes to the margin of error and the point estimator.

Otherwise, everything is just the same, alright? So we've seen these types of problems before. Let's go ahead and get started. Now, when you're creating a confidence interval for matched pairs, remember, you're always focusing on the difference between two values.

So your confidence interval is going to be for the population difference, that's μ_d, and then you're creating an interval by using the sample mean difference as a point estimator. That's, like, your best guess. And then we'll talk about this margin of error in just a second. Alright? So we've already seen this problem.

We already know the result of it. Let's go ahead and take a look here. We have a study on the effectiveness of new blood pressure medication, and the sample mean difference in thirty-seven patients' blood pressure is taken before and fifteen days after. And the sample mean difference is 3.2 with a standard deviation of 6.7. So in other words, we know we have that n, the number of pairs, is 37.

We know that the ̄_d, the sample mean difference, is 3.2. And then we have SD, the standard deviation of the difference, is equal to 6.7. Alright? So we already have all of those values, and we've seen them before. We just pulled them out of the problem.

So we're going to create a 90% confidence interval. Similarly to other types of problems we've seen before, we're going to use this result or this interval to make an inference about a claim that's being made, in other words, a hypothesis test. Alright?

First, we're just going to verify a few basic conditions. We need matched pairs, and we need them to be randomly selected. So presumably, we're just going to take 37 randomly sampled patients, and they're going to take their medication. We can assume that's true. And we know that this is a before and after.

There's a one-to-one pairing of each value in the dataset. These definitely are matched pairs. Next, we need it to be a good sample size, or we can assume normal population distributions. But in our case, we have n greater than 30, so that condition is met.

Next, we're going to find a critical value from our confidence level, which is 0.9. We're going to find a critical t α/2.

This comes directly from our confidence level. Now, remember, with these types of confidence intervals, we always use the t rather than the z. We are going to use a t-score, critical t-score. This just comes from taking one minus that number and dividing it by two (t = 0.05).

With a z-score, a z 0.05, this is always 1.645. But with a t-distribution, you have to use it with the degrees of freedom to find this critical value. The degrees of freedom here work the same way as any other type of test; you subtract one from the sample size, so this is going to be 36.

Using a t-distribution with degrees of freedom of 36 to find the critical value, you will see this is 1.688. For a normal distribution, a z-score, this is 1.645, so this is pretty close.

Now, we're going to use our point estimator, which is our best guess at the middle of our interval. Sometimes we must calculate these, but in our case, we already have what ̄_d is equal to because we just wrote it down over here; ̄_d is equal to 3.2. We don't have to calculate it; just pull it right out of the problem.

Now, we just need to find our margin of error. We have our best guess, our initial guess, for the middle of the interval. The margin of error equation for matched pairs is straightforward. It's just the critical t-value, and then you multiply by the standard deviation of the difference, SD, over the square root of n.

Our margin of error here is going to be our critical t-value, 1.688 times the standard deviation, which is 6.7, divided by the square root of 37. If you multiply this, which is easy to stick into your calculator, you should find a margin of error of 1.85.

Now that we have our margin of error, remember, we always have our point estimate and our margin of error, and then we build our upper and lower bounds by subtracting and adding that margin of error. Basically, our interval will be 3.2 minus 1.85, and we'll add zero there just to make it two decimal places. That'll be the lower bound, and for the upper bound, we'll have 3.2 plus 1.85.

Our interval will be from 1.35 on the lower end to 5.05 on the upper. This is your 90% confidence interval. And that's all there is to it; we've seen these types of problems before. It's not a whole lot different for matched pairs.

Now let's take a look at the second part of our question because we've seen this type of language before. What does this result suggest about the claim that there is no difference after taking the medication? This is a hypothesis test.

What does our confidence interval tell us about that hypothesis test? The rule is the same for matched pair confidence intervals. You're able to look at whether your interval includes or doesn't include the number zero. So we are 90% confident that the true difference in blood pressure before and after taking the medication is between 1.35 and 5.05. Since this interval does not include zero, both numbers are positive, we reject the null hypothesis, which is that the difference in mean for the population is zero. There is evidence that there is a significant difference in blood pressure before and after.

This is how it works for matched pairs and how we link confidence intervals back to a hypothesis test.

Thanks for watching. Let's go ahead and take a look at a couple of practice problems.

Construct a 95% confidence interval for the mean difference of the population given the following information. Would you reject or fail to reject the claim that there is no difference in the mean?
$\bar{d}=-0.728$
$s_d=1.34$
$n=10$

• A. The 95% confidence interval = $(-1.152,-0.304)$; Fail to reject the claim that there is no difference in the mean.
• B. The 95% confidence interval = $(-1.686,0.230)$; Fail to reject the claim that there is no difference in the mean.
• C. The 95% confidence interval = $\left(-1.686,0.230\right)$; Reject the claim that there is no difference in the mean.
• D. The 95% confidence interval = $(-1.152,-0.304)$; Reject the claim that there is no difference in the mean.

Answer: (B)

Here’s what students ask on this topic:

What are matched pairs in statistics, and how do they differ from independent samples?

Matched pairs in statistics refer to samples that are related in a unique way, often through a before-and-after comparison of the same individual or related individuals. This relationship allows for a one-to-one pairing of data points between the two samples. In contrast, independent samples are not related and do not influence each other. The key difference lies in the dependency of the samples; matched pairs are dependent, while independent samples are not. This distinction is crucial when conducting hypothesis tests, as it affects the statistical methods used. For matched pairs, the analysis focuses on the differences between paired observations, whereas independent samples are analyzed separately.

How do you conduct a hypothesis test with matched pairs?

To conduct a hypothesis test with matched pairs, follow these steps: First, identify the paired samples and calculate the differences between each pair. The null hypothesis typically states that the mean difference ( $\mu$_d ) is zero, indicating no change. The alternative hypothesis suggests a significant difference. Calculate the sample mean difference ( $d$_bar ) and the standard deviation of the differences ( $s$_d ). Use the t-distribution to find the test statistic, replacing variables like $x$_bar with $d$_bar . Compare the p-value to the significance level ( $\alpha$ ) to determine if you reject the null hypothesis. This method allows for robust analysis of dependent samples.

What is the formula for calculating the mean difference in matched pairs?

The formula for calculating the mean difference in matched pairs is given by the sample mean difference ( $d$_bar ). It is calculated as follows:

$d$_bar = ∑ d i n

where $d_i$ represents the difference for each pair, and $n$ is the number of pairs. This formula provides the average difference between the paired observations, which is essential for hypothesis testing and confidence interval calculations in matched pairs analysis.

How do you create a confidence interval for the mean difference in matched pairs?

To create a confidence interval for the mean difference in matched pairs, follow these steps: First, calculate the sample mean difference ( $d$_bar ) and the standard deviation of the differences ( $s$_d ). Determine the critical t-value ( $t$_α/2 ) based on the desired confidence level and degrees of freedom ( $n-1$ ). The confidence interval is then calculated as:

$d$_bar ± t _α/2 × s _d / n

This interval provides a range within which the true mean difference is likely to fall, offering insights into the significance of the observed differences.