����app

Welcome back, everyone. So in previous videos, we've talked a lot about how to do a hypothesis test of means when we're given one sample. But as you get deeper into the course, you may start to run across some problems where they're going to give you two samples instead of one and ask you to do a hypothesis test about the difference in means between the two groups. Now this might seem like it's going to be twice the amount of work or twice as complicated, but, thankfully, all of the basic steps are all the same. We're going to write some initial hypotheses, calculate some test statistics, find p-values, and then write our conclusion.

Now there's a few important differences that we're going to talk about here, so let's just jump right into our problem, and I'll show you how all this works. Alright? Let's get started. Now the basic difference with a hypothesis test where we're given two samples is basically instead of focusing on one specific mean equaling some number, our claims that we're testing are always going to be about the difference between the two sample means. Let's take a look at our problem here.

So, this table summarizes a study that's done on the mean resting heart rate between males and females. We're given the sample size, means, and sample standard deviations. Now we're going to perform a hypothesis test, the level of significance that's already given to us, and we're going to do a hypothesis test about the difference in means between the two groups, and we're going to assume normal population distributions. Alright? Now, by the way, if you've seen our video on two proportions, a lot of the setup is going to be very, very similar to that.

If not, that's perfectly fine. Now remember, the first step is always to write an initial hypothesis. For a one-sample mean test, we always just found a number, and basically that was coming from the problem, and we said that mu is equal to that number. That was the default assumption, like, mu is 45 or a hundred or something like that. So how does it work now when we have two means instead of one?

Well, again, our claims are always going to be about the difference between the means. These problems are always going to be set up this way where there's going to be a difference. And so what you're going to do here is you're going to write your initial hypothesis as mu one is equal to mu two. Your default assumption here is that the two means are exactly the same. This is always how you're going to write your initial, h0.

Alright? So another way of writing this that you may see, by the way, is that the difference mu one minus mu two will equal zero, and we're going to see why you can write that in just a second here. Alright? That's basically always going to be your first step, that HO, which is that mu one is equal to mu two. That's your default assumption, that the two means are exactly the same.

So your alternative hypothesis, mu one and mu two, you're always just still going to have to figure out the symbol that goes here, less than, greater than, or not equal to. And in this case, we don't really have any indication that it's a less than or greater than, so we're just going to go ahead and default to a not equal to. By the way, this just means that it is a two-tailed test, and we're going to see later on why we're going to need that when we calculate the p-value. Alright? Now before we get started with the rest of the problem by calculating the test statistic, there's a couple of conditions that we have to check for very quickly.

We have to check that these samples are random and independent. In a lot of cases, you can kind of just assume that, but in this case, we're told explicitly that they are random and the two groups don't interact with each other in any way, so they're independent. Now the second thing is really, really important. We have to basically know or we have to actually assume that the population standard deviations, sigma one and sigma two, are unknown, and they're not assumed to be equal. In these types of problems, they will always give you the sample standard deviations, and you're going to assume that you don't know what the population standard deviations are.

We'll show you in later videos how to deal with these types of situations where that isn't the case. Alright? Now the last thing you check for is that the both samples are normal or you need to have big sample sizes. In this case, our sample sizes are small, but we're going to assume normal population distributions so that condition checks off. Alright?

So now the second thing we have to do is we're going to have to calculate our test statistic. Alright? Now when we did this before when sigma was unknown, we always use the t-distribution. We're going to use the exact same thing for a two means, t-test. So this is sometimes referred to as a two means, two-sample t-test.

Now, basically, the basic setup is all the same. We're still going to have a sample mean, except now we're just going to use a difference in the sample means, and then we still had a minus a population mean. In this case, we're going to use the difference in population means, and then we had something in a square root, which is like a standard deviation down here. Alright? So let's just take it one term at a time.

So our first term in our t equation is going to be the difference in the sample means. Now one of the things I always like to do in these problems before I even get started with that is a lot of times in these problems, the information won't be as organized as it is in this table over here, and they'll kind of be just jumbled up inside of a paragraph. So what I would like to do is always extract the numbers and just have them in two really neat columns. So for males, which is group one, and females is group two, I'm just going to basically read off numbers from the table here. My sample size is 10.

My sample mean is 70.2, and my sample standard deviation is 5.8. For females, it was 11, and I had 81.4 and then 6.4.

So in a lot of cases, you may not be given much information about the setup of a problem. You may just be given a bunch of numbers and asked to sort through the steps of a lengthy process like a hypothesis test. That's exactly what's going on here in this example problem, so we're going to work it out together. We're going to use the information below to test a claim that's being made about a difference between two population means at some specified significance level. We have α=0.1

So, again, we know nothing about this problem. We have no idea what these numbers represent. That's very often the type of problem you may actually run into. What's nice about this is because you just get to focus on the numbers and how they work inside the equations, but let's get started here. We're going to assume the samples are random and independent.

It's always something we check for. We assume normally distributed populations, and we have unknown and unequal population standard deviations. This just takes care of a bunch of different assumptions that we make at the start of these problems. Right? So remember, what's the first thing you do when you're supposed to test a claim or use a hypothesis test?

The first thing you always have to do is write your null and alternative hypotheses after you've checked a bunch of conditions, but we've already checked those. Right? Your H0 and your that's where you start. Alright? Now in our case, the claim that's being made here is that μ1>μ2.

Now, that's just a claim that's being made that's not necessarily your null hypothesis or alternative hypothesis. You have to turn it into your H0:μ1=μ2. Alright? Now, your Ha1:μ1>μ2 is where you start to include some of those symbols in there, and as we can see, our claim is that one is greater than the other.

By the way, this helps set up the next parts of the problem, which is when we calculate the t and the p-value, this is going to be a right-tailed test. This is something you know right off the bat. Okay?

So, let's go ahead and use the and move on to the second step. Alright? In the second step, this is where we use all of the numbers and actually calculate a test statistic. In this case, because we have sigmas that are unknown, this always ends up being a t-score.

Alright? So we're going to use all of the values that we have in here in this equation to set up our t-score. Now if you need to go refresh and look at the equation for that t-score, go ahead and do so. Basically, this t-value is t=(x̄1−x̄2−0)s1/n1+s2/n2, giving a value of 1.359.

Now, the probability of our sample being as unusual as it was, the p-value, we calculate that next. We'll need the degrees of freedom, which are 18 in this case. Using a t-score of 1.359, the t-score probability is 0.0955. Remember, since p is less than α, we reject the null hypothesis because there does appear to be a difference in these means (sufficient evidence).

So that's really all there is to it. That's steps one through four of a hypothesis test with two samples. It's very similar to how it worked with one. Let me know if you have any questions.

Thanks for watching.

So we just finished learning about how to conduct a hypothesis test of means when we're given two samples instead of one. Now, what you may start to see in these problems is they may also ask you to construct a confidence interval for the difference in the means between these two samples. I know it sounds kind of complicated, and we're going back to confidence intervals, but don't worry. We already know all of the steps for making confidence intervals for one mean. Now we're just going to do it for a difference.

Really, all the steps are pretty much the same. We're just going to need some slight modifications for our point estimator and our margin of error. Otherwise, everything is exactly the same. Alright? So let's just jump right into this problem here, and I'll show you how this all works.

Alright. So the basic difference between these types of problems with two samples instead of one is that your confidence interval is going to be made about the difference in the population means by basically using the sample means. We're going to be using our point estimator that's the difference between the sample means and then just a slightly different margin of error equation, which we'll get into in just a sec. Alright? So in our example, let's go ahead and take a look.

We've already seen all of these numbers before. It's a study on the mean resting heart rate of randomly sampled males and females. We have all the sample data over here, the size, the standard deviation, and things like that. We're going to construct a n90% confidence interval for the difference, and we're going to assume normal population distributions. Now, the second part of this problem here, we're going to get to in a little bit, which basically is about a claim that's being made.

We'll talk about that in the second half of the video. Let's jump right into the steps here because we already know how to do all this. First thing we have to do is just check that these samples are random and independent, which we can almost always assume here. We have randomly sampled males and females, and they're independent because they don't interact with each other.

Now we have just assumed that the samples come from a normally distributed population or that these sample sizes are big enough. In our case, the sample sizes are small, but we can assume normal population distributions, so we have that condition as checked. Let's move on to the second step over here.

We're going to find the critical value. So remember, that's tα2. This comes from your confidence level. Our confidence level over here is 90%, so 0.9. This basically gives us what our t-value is.

This essentially means that in a two-tailed sort of distribution, remember, you're going to be looking for what's the t-value that gives you 5% of the distribution on each side. And remember, now for a z-value, for a z-distribution or normal distribution, that t-score was always the same. For a t-distribution, it's a little bit different because it depends on the degrees of freedom, and the rule is exactly the same as it was for a hypothesis test. You're always just going to use the degrees of freedom that corresponds to whatever the smallest sample was in your sample data. So in this case, it's just the males, which has a sample size of 10.

Therefore, your degrees of freedom is just going to be nine. This is always how you're going to do this. Alright? It's the same exact rule that you just learned for the hypothesis test. So if you go ahead and look this up here over here, this is going to be a t-score of 1.833t.

Alright? So that's the first couple of steps here, steps one and two. Now remember, we're just going to have to go ahead and figure out our point estimator. What is the best guess for the difference in sample means as a starting point? Remember, that's also the middle of their confidence interval.

Alright? So this is just going to be x1-x2. Now when you have the data that's kind of organized like this nice and neatly here, this is really, really straightforward because you just pull it straight from these numbers. We actually are already given what the sample means are, so you just straight up subtract those two numbers. So this is going to be 70.2-81.4 for a sample difference of -11.220.

Alright? So this is going to be your difference in sample means. Now let's move on to the fourth step over here. That's your best guess for the difference in sample data, but now we're going to have to go ahead and calculate the margin of error. Right?

So the margin of error is going to be this equation over here that relies on t-score, and there's a slight modification that happens inside of the square roots. Now we have two samples instead of one, so the only thing we could do here is we just have s12+s22 over n1+n2. Right? There you go. So you basically just use both of these standard deviations, always the ones with the ones, the twos with the twos, and just follow this equation here.

Alright? So in other words, our margin of error is just going to be the t-critical that we just calculated, the 1.833. So 1.833×s15.82÷n1,ns26.42÷11. Alright? Now you're going to go ahead and work this out, and you're going to find a margin of error in your calculator, just be really careful when you do this, of 4.88.

Right? That's your margin of error in your confidence interval. Alright? Now we just put all of these things together to figure out what are the upper and lower bounds. This works exactly the same way.

You're just going to take your point estimator, subtract e, and add e to get your upper and lower bounds. So for our very last step, what we're going to do is take our point estimate that we just got from step three, and this is going to be -11.2-4.88. That's our lower bound. And then we're going to take 11.82 and add 4.88.

That's going to be our upper bounds. And when you work this out, what you're going to find here is that this is going to be -16.08 for our lower bound, and our upper bound is going to be -6.32. Alright? So both of these are negative numbers. And this is our 90%CI. Alright? That's the answer. So this is what you would do. Now remember, in some courses which may have some stricter requirements, you may be asked to sort of write out a little paragraph that summarizes all of this so we can do that really quickly here.

So, basically, what this means here is we're 90% confident that the true difference in mean resting heart rate between males and females as a population is between these two numbers over here. So we have -16.08 and -6.32. So they're both negative numbers. Alright?

Now, so what we're going to do is we're going to go back to the second question over here that I promised we would get back to. So what does this result suggest about the claim that there is no difference in mean resting heart rate between males and females? This is what's going to happen to a lot of these problems. They'll ask you to make a confidence interval, and in the second part, they'll ask you to use it to make a determination about some claim that's being made. In other words, a hypothesis test that's being done.

Now here's the connection. Basically, what we always started with hypothesis testing was that our H0, the initial hypothesis, was that the two means were equal to each other. In other words, another way of writing this is that μ1-μ2 was equal to zero. There is no difference in the mean. That was our default assumption.

So what's going to happen is in our confidence intervals, we're always going to be looking for this number over here, zero. This is basically going to determine whether you're going to reject or fail to reject. The rule here is as follows. If your confidence interval does not include the number zero, then what you're saying is that you're ninety-first percent confident that there actually is a significant difference between these two means, between μ1 and μ2. Because there is a difference, what you're going to do is you're going to reject that null hypothesis because your null hypothesis is that there is no difference.

So you are opposite that, you're going to reject that. However, if your confidence interval does include zero, then it's the opposite. It's possible that there's actually no difference between these two sample means or two population means. Therefore, you're just going to fail to reject. It's a little bit of a double negative, but if it does include zero, that means that it's possible that there is no difference, so you're going to fail to reject it.

Now going back to our problem here, we can see that we have an interval with two negative numbers, negative 16 to an upper bound of negative 6.32. Alright? Now because this does not include sorry. Because this does not include zero, then what we're going to do is we're going to reject the null hypothesis that they are actually equal to each other. In other words, there is enough evidence that there is actually a difference in mean resting heart rates between these two samples.

As we actually saw in our first video on hypothesis test, we saw that we rejected the hypothesis test, and that's exactly what we found here as well, so those are consistent. Alright, folks. That's really all there is to it. Thanks for watching. Let's get some practice.