10. Hypothesis Testing for Two Samples
Two Means - Unknown, Unequal Variance
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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 or the difference in the means between these two samples. Now I know it sounds kinda 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 gonna do it for a difference. Really, all the steps are pretty much the same. We're just gonna 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 gonna be made about the difference in the population means by basically using the sample means. We're gonna 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, things like that. We're gonna construct a 90% confidence interval for the difference, and we're gonna assume normal population distributions. Now the second part of this problem here, we're gonna 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 just jump right into the steps here because we already know how to do all this. Alright? 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 things because they don't interact with each other. Alright? Now we have just assumed that the samples, are 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. Alright? Let's move on to the second step over here. We're gonna find the critical value. So remember, that's t alpha over two. 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 is basically just means that in a two tailed sort of distribution, remember, you're gonna 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 gonna 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 gonna be nine. This is always how you're gonna do this. Alright? It's the same exact rule that you just learned for hypothesis test. So if you go ahead and look this up here over here, this is gonna be a t score of of 1.83 or sorry, 1.833. Alright? So that's the first couple of steps here, steps one and two. Now remember, we're just gonna 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 gonna be x one minus x two. Now when you have the data that's kinda 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 gonna be 70.2 minus 81.4 for a, for a sample difference of negative 11.2, two zero. Alright? So this is gonna 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 gonna have to go ahead and calculate the margin of error. Right? So the margin of error is gonna 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 s one squared plus s two squared over n one plus n two. 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 gonna be the t critical that we just calculated, the 1.83. So 1.833 times, and in square roots, we're gonna have s one. Remember, s one in these types of problems, you're always given s one because sigma for both of them is assumed to be unknown and not equal to each other. It's always how it works in these problems. This is gonna be 5.8 squared over the sample size n one, which is 10. So this is gonna be, you know, n one, and this is gonna be n two and so on and so forth. And then you're gonna have 6.4 squared, over, and this is gonna be 11. Alright? Now you're gonna go ahead and work this out, and you're gonna find a margin of error in your calculator, just be really careful when you do this, of 4.88. Right? That's so 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 gonna take your point estimator, subtract e, and add e to get your upper and lower bounds. Alright? So for our very last step, what we're gonna do is take our point estimate that we just got from step three, and this is gonna be negative 11.2 minus 4.88. That's our lower bound. And then we're gonna take 11.82 and add 4.88. That's gonna be our upper bounds. And when you're gonna work this out, what you're gonna find here is that this is gonna be negative 16 o eight for our lower bound, and our upper bound is gonna be negative 6.32. Alright? So both of these are negative numbers. And this is our 90% confidence interval or the difference in the two population means. So this is 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. 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 negative 16 o eight and negative 16, 6.32. So they're both negative numbers. Alright? Now so what we're gonna do is we're gonna 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 gonna 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 with hypothesis testing was that our HO, the initial hypothesis, was that the two means were equal to each other. In other words, another way of writing this is that mu one minus mu two was equal to zero. There is no difference in the mean. That was our default assumption. So what's gonna happen is in our confidence intervals, we're always gonna be looking for this number over here, zero. This is basically gonna determine whether you're gonna 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 mu one and mu two. Because there is a difference, what you're gonna do is you're gonna reject that null hypothesis because your null hypothesis is that there is no difference. So you are opposite that, you're gonna 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 gonna fail to reject it. Now going back to our problem here, we can see that we have a 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 gonna 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.
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