there we go there we go okay so right good So last time we had started to talk about inferential statistics and we set up an analogy of how to understand inferential statistics. Anybody happen to remember how we were thinking about inferential statistics? There was a particular job, a very well-paying job.
Lawyers. Now, being a lawyer. So the whole idea of this analogy, and it's a correct analogy, so in other words, there's not an obvious fault with anything about it, is that to be a lawyer, you're making an argumentation that something is correct. Well, that's what the idea of inferential statistics is, as the definition that's there is inferential statistics involves making. predictions or inferences about a population based upon a sample of data drawn from the population.
So that's the same thing, exactly the same thing, as what takes place in a court of law. So the samples are the collection of information, and then that collection of information is what is used to be able to make statements about the population. So then we have then what we refer to as sampling and sample.
distributions. So sampling is the process of selecting a subset of the total population from which we are then going to estimate characteristics of the entire population. So the idea of this is that suppose that we were taking the entire population of individuals in the United States, and then we were then going to say, okay, well, we're going to pick 100 people from each state.
And we're going to randomly select these individuals from each of the 50 states. And we're then going to try to then make statements about the general population of individuals in the United States based upon that. So if then when we have collected this information from the 100 individuals in each of the 50 states, so 100 from Pennsylvania, 100 from Ohio, 100 from West Virginia, 100 from Virginia, 100 from Maryland, et cetera, Suppose that out of that group, we found that almost like 75% of the people that we sampled hate some particular poster that we came up to advertise our product. Well, if 75% of the individuals that we had sampled hate the poster we came up with, then we would probably not want to launch that poster as a part of a marketing campaign. And so that's the kind of example of how we would utilize sampling as a technique.
And so the idea is, is we can't ask everybody in the country, but we can, in fact, ask a subset. So here we have a different business example. So a marketing firm is trying to gauge public opinion on a new product, very similar to what I just said.
Instead of surveying the entire population, they randomly select 1,000 individuals to survey, which is less than the amount I was saying. The average opinion score from this sample can be used to estimate the overall public opinion, which is correct. We can do that.
Additionally, the firm can create a sampling distribution by repeatedly drawing samples and calculating the mean opinion score for each sample, which provides insight into the viability of their estimates. And given here are some ideas of how to do those things using Excel, we will get back to that. in particular when we go down to a further subsection of this guide.
So we are eventually going to go into this guide, which has some sample data sets for us to use for testing some of these things. So we're currently working in the section that kind of just overall explains these concepts, and then we'll go down to then some more specific things in a few moments. So confidence intervals. So confidence intervals are something that are very commonly misunderstood. And the reason why they're very commonly misunderstood isn't because they're particularly difficult.
It's because of the fact that it's usually not thoroughly explained what the concept of inferential statistics is, period. And therefore, it's very difficult to understand what you're supposed to be getting out of these things called confidence intervals. So a confidence interval is a range of values by which we can be confident in making a statement about a parameter. Okay, that's a very technical sounding definition. Let's make it more practical.
Okay, so in this room of individuals, do you suppose there is an average height? Yeah, there has to be, right? Because we have a group of individuals.
Okay, well now, we're not going to make everybody stand up. And thus, it makes it much more challenging for us to make a guess about what the average height is. Now, we can draw some conclusions based upon national averages. And we can also say, well, we know what the national average is for men.
And we know what the national average is for women. And we know how many men versus women are in the room, so we can draw some pretty good conclusions based upon that. But we can also, though, say we also have some things that are skewing the data a little bit.
So anybody in here happen to play any sports for Seton Hill? OK, and what sports do you happen to play? OK, so for the sports that are represented.
Do you suppose that it is correct that those sports that are represented feature individuals who tend to be not tremendously above, but a little bit above the national average in height? The answer is yes. Okay. So already we know that just the national average data probably shouldn't work super, super well.
Okay. So that means that our confidence how well we are, how strongly we are representing information should go down a little bit if we were to just list the value that's the national average. Now, what is the national average for men? You can Google that right now.
What is Google saying the national height, national average height for men is? I think it used to be 5'9". I don't know if it still is. Still 5'9"? Okay. So That's the national average for height. Well, so then we then say, okay, well, what if we come up with a range of values?
And in particular, what if I told you I was 90% confident that the height should be of the average height of individuals in this room for both men and women should be somewhere between 5'3 and 6'1. Sounds okay, right? Now, I'm completely making this up. But if that were true, then let's investigate what the meaning of that would be. So by 90% confident, I'm saying that there's a percentage chance I'm wrong.
And the higher the confidence, then the stronger the likelihood that I'm getting my estimate. correct. Now, in particular, I'm not stating it as a single number. I'm stating it as what we refer to as an interval range.
So I said 5-3 to 6-1, right? I think that's what I said. I feel pretty confident that the average number of individuals in this room is somewhere in between 5-3 and 6-1.
Now, a way that I could become more confident is to have a larger range. So for instance, if I said I think that the average height of individuals in this room is in between 5 feet and 7 feet, I would feel much more confident. I would feel amazingly confident. And in fact, With that information, I would feel confident to go into basically any room on the planet and have that as at least my initial guess. Five feet to seven feet.
average height. There are really very few places where that's going to be a bad interval estimate, right? What's one of the few places on the planet where if you had that as your guess at the average height, you'd just be bad?
Elementary school. If you go into any elementary school, assuming you have the appearances, if you go into any elementary school, If your guess for the average height is five feet to seven feet, you're going to be wrong. It's not going to happen.
Another way that you could do that would be to go where? The locker room for certain teams of the NBA. you have a possibility that potentially that might be a bad estimate on a particular given day. Okay, so the idea of an interval range like this is that we would like to be able to have it so that our confidence level states how sure we are that our... parameter, and our parameter is the thing that we're trying to estimate is in between the numbers that are given.
Now, there's a set way to come up with these things called confidence intervals, and we state them in percentages. Now, certain percentages are given for these things for traditional reasons, and I'll talk about that. what I mean by these traditional reasons.
The percentages that are most often given for the confidence are 90, 95, 98, and 99% confidence. Now, there is nothing special about those numbers, but those are the standard numbers that are given. Could you give someone an answer about 96% confidence? You 100% could, and it would be more meaningful than 95% confidence and less meaningful than 99% confidence, but 96 is never what people are given.
Okay, why is it that these four numbers that I stated, which are 90%, 95%, 98%, and 99%, why is it that they're magical? Well, the answer is, is that they're not magical. And it's just that back in the day, and what do I mean by back in the day? Well, back like in the particular, the 50s through the 90s, to be able to do these calculations was time intensive. We live in a very special age.
We live in the age of data, which is past the age of computers. It's still a part, you know, we're still in the computer age and that everybody has a computer. It's just that in terms of what things are labeled, it was like the, you know, the 80s through early 2000s, they were labeled as the computer age.
And really the reason why they were labeled as the computer age is because that's when having a personal computer became a thing. Computers have been around since, honestly, the late 1800s. But computers as we know them, like the ones that are in front of all of you, those didn't exist until the 80s.
Now, they were much larger in the 80s than the ones that are in front of you. But that's what we'd refer to as a personal computer. It's a computer that you as an individual own. How many have seen what computers looked like in the 30s, 40s, 50s, and 60s? Okay, so Google this right now.
this is a very important thing type 1950s computer and is there a picture of this okay so what kind of pictures are you getting so i'm going to come around so everybody have everything okay Okay, so this kind of idea of the whole room thing, like that console, like that one right there. That's what we're talking about. Yeah, that wall, the guy's wall looks up.
So we're talking about things that are taking up entire rooms. Now, here's the thing. The amount of computation.
Yeah, that's actually exactly right on the spot. So the amount of computations that those can do. The ones that you're seeing that take up a whole. desk.
I'm going to show you a device that's more powerful than that. Are you ready for it? Okay, this is a very silly keychain. This is my, if someone says something that I don't 100% think is necessarily as worthy of complaint as they think it is. I will tell you, I will never use that on a student.
I only use that on co-workers. But anyways, inside of that little device, there's a tiny little microchip. The power of that microchip exceeds the power of the desk-sized computer that you saw when you pulled up 1950s computer.
The one that's the whole room computer is not as powerful as the calculator's. that you used when you were in elementary school. When you were in elementary school, you were using a calculator that was more powerful than those entire room computers.
Now, in the modern era, an entire room computer is the kind of computer that is used dedicated for things like nuclear power plants. Back then, entire room computers for what we're required to run basic calculations. So we've gotten much better as a society. Okay, so going back to these percentages that I listed, for a very long time to do a calculation like what we're going to do was a very time-intensive task, very time-intensive. So any of the calculations we've done so far would have been something that would have, that people either would have done by hand, or if they wanted to use a computer, they would have had to put in a request for the usage of a laboratory space.
They would have had to have used what's referred to as a punch card, which is like a piece of cardboard that you have, that it has holes in it, that you input into the computer, that the computer will then read from, because There weren't things like flash drives. There wasn't an internet of where you could upload a file or anything like that. It was all of these extremely archaic features.
And so as a result, there were certain percentages for confidence intervals that were selected as the ones that people would use to standardize calculations so that they could be written down in books because most people would not have access to a computer. So inside of a textbook would be a list of table values for the percentages 90, 95, 98, and 99. So the only reason those numbers are still used today is because within the realm of business analytics, those were the numbers that used to be used. And so by tradition, those are the numbers that are still used. Okay, so how does one do this thing called a confidence interval? Well, there's a step-by-step process, which is that the first step for doing a 95% confidence interval is that you find what's referred to as the standard error.
We will actually go through an example in a bit. At the moment, I'm going to tell you what I want you to take out from when I'm going through this initial treatment of these topics. All I want you to do is to take away the big idea picture of what I'm talking about. So I haven't done actual examples on any of this yet.
So all I want is for some of these terms to start to get into your head. Does that make sense what the goal is? So if you're thinking in the middle of my current talking, wow, I don't know how to do any of this, and I'm not learning how to do any of this at the moment, I agree with you.
You're not learning how to do this. And that's because all I want is... some of the language of it. to be what you become familiar with. There's a big difference between the language and the skill.
These terms and words are things that I suspect are very new to all of you. Is that correct? Are all these new terms like things like inferential statistics, confidence interval, et cetera, et cetera. Okay. So I'm wanting these terms to become familiar enough of where when I then come back to them to start going through examples, then you know what some of the words kind of mean.
Does that make sense? Okay, so then we want to do the calculation of the standard error, which to compute the standard error in Excel, it's given right there. And then we would determine what's referred to as the critical value. We will come back to what that means.
And then we multiply the standard error by the critical value to find the margin of error. We will again come back to talk about all of this. but that does give the complete breakdown.
Now, hypothesis testing, and all of these topics here are all connected, by the way, which is the reason why I'm giving kind of an overview of them all at once, because there's going to be a, there's a similar flavor to all of these notions. And that similar flavor is the concept of making an argument. Again, just that notion. It's like we're lawyers. It's making an argument.
So the next thing that we come to is what's referred to as hypothesis testing. And hypothesis testing is what most people think of when they think of inferential statistics. So what is a hypothesis?
Now, like a theory is how it's typically understood, but it's actually even easier than that. Okay. A hypothesis is just a claim about something. Now, a hypothesis can be true or it can be false.
So let me make a hypothesis. So here's an example of a hypothesis. Bigfoot exists in Pennsylvania. That's a hypothesis.
Now, is it true? Maybe, maybe not. Is there a truth value to it?
Is it either true or false? Well, the answer is yes. It's one of those two things.
Now, is there any way for me to prove it? The answer is yes. What's the only way I could prove that?
that Bigfoot exists in Pennsylvania. If I capture Bigfoot, it is truly, and in particular, if I capture Bigfoot in Pennsylvania, that is the only way for me to prove that statement, right? Why is it that taking a picture will not be enough?
Could be fake. So only way to prove Bigfoot exists in Pennsylvania is for me to capture a live Bigfoot. A live Bigfoot needs to be alive. Because if it's dead, nothing says it came from Pennsylvania.
I need to have video of me live capturing in the wild Bigfoot. That's it. It's the only absolute way. And you probably didn't wake up thinking you were going to learn about how to, you know, capturing Bigfoot in your quantitative methods class.
Okay. But so the idea of a hypothesis is it has to be true or false, but it doesn't mean that you necessarily know whether it's true or false, but it has to be true or false. So what's another example of a hypothesis?
Today is Monday. That's a claim. Now, is that one true?
That one is true. And we know that it's true. Why? Well, because all of our calendars say it.
So that one's true. And we have the reason why. Now, let's do another example of a hypothesis. Coca-Cola is the best cola. Now I'll tell you that one's a trick.
It is true or false, but it's truth or falseness depends upon what your definition of best is. If your idea of best is tastes the most like Pepsi, then Coca-Cola is not going to be the one. So it does have a truth value, but it doesn't mean it's necessarily immediate and instant what its truth value is.
Okay, now the idea of hypothesis testing, as it says, is a statistical method used to make decisions about the population based upon sample data. So when we do hypothesis testing in statistics and quantitative methods, what we're doing is that we're testing a claim to find out whether it's true or not. And the only way that we can test the claim is to gather information via samples. And I do give an example here of where if you happen to have an Excel file, how to go through that.
The next term that I'd like to start getting kind of a little bit into the conversation are what are called p-values. And p-values are short for probability values. And these things called p-values are used extensively in hypothesis testing. And the reason why they are is because when I make a claim about something, there's some probability likelihood that I have correctly described the claim in terms of its truth value. So again, let's go with that example of in terms of Bigfoot in Pennsylvania.
Okay, now I will tell you in advance, I think it would be awesome if Bigfoot existed. That would make my day. If tomorrow we all woke up, there was a press release, all the leaders of our country came out and said, gotta tell you, we didn't think we would be united on anything, but we've discovered Bigfoot, and it's brought peace to our country.
And in fact, all wars throughout the world are currently ending, because we now realize. there's a greater cause to human beings. Like, man, that would be awesome.
But I'll also tell you, I think it's amazingly unlikely. So I'm going to say that I think, in fact, the claim Bigfoot exists in Pennsylvania, I think it's false. It's not that I want it to be false.
It's that I think it's false. And so then we're then going to say. Well, that claim, therefore, we're going to label as untrue. And then any kind of then result of information that we collect.
So, for instance, you know, if we put up trail cameras and they come back empty, then we then say, OK, well, for every trail camera that comes up empty from not having observed Bigfoot in the wild, it's in fact continuing to validate. the falseness of the claim that Bigfoot exists in Pennsylvania. And then that leaves us then with the fact that there's some probability that that claim is false. And we probably happen to believe that the probability that that claim is false is very, very high, even though we might want in our hearts it to be true.
Well, that means that this thing called probability values, p-values, are connected to this idea of hypothesis testing. And again, we're going to see more details about this. But the big thing that I want you to have as a takeaway is that p-values are a way of talking about the strength of something.
In particular, they have to do with the strength of evidence. against the null hypothesis. And that's a very important phrasing.
It's the strength of the probability of the evidence against the null hypothesis. Now, where is that coming from that that's the thing that we're going to care about? Well, let's remember we said we're statistical lawyers, but we're statistical lawyers of a very particular type. We're for the prosecution. And in a court of law, it's innocent until proven guilty.
So innocent is the claim that stands. And then if we are lawyers for the prosecution, then we're trying to come up with evidence that's against the innocence. So for instance, if there is a claim.
that there's no lead in the water in some creek running through Seton Hill property, there's no lead in the creek, then as good researchers, what we should be doing is we should be trying to find lead in the water because we're looking for evidence against the claim. And then at the end of the day, if there's no evidence of there being lead, then what is it that we're left with? We're left with the conclusion that there's no lead in the water. Okay, so that's the way that hypothesis testing and p-values work.
It's that we're going to make a claim, we're going to look for evidence. And then there's going to be then something that we call a p-value, which is the likelihood of us successfully defeating the claim. Cool. Now, as a part of that, there are then two testing procedures for testing hypotheses that are going to be particularly important for us.
What are called z-tests and t-tests, both of which are going to be very important. Now a z-test and a t-test use two different distributions. The z-test uses what we already learned about, which is referred to as the normal distribution, and the t-test uses a very similar distribution, which is what's referred to as the t-distribution.
Now there's an additional name to the t-distribution that I'm also going to teach you. And by teaching you the history of this different name for the t-distribution, it's going to give us an example of how these concepts are used practically in a business scenario. So an alternate name for a t-distribution is what's referred to as the student t-distribution. And there's a reason why it's called the student t-distribution. It's not that there's a secret professor t-distribution, and therefore then there's also a student t-distribution.
It actually comes from a very interesting story from history. And it's a true story from history, which is what makes it even more interesting. So towards the end of the 1800s and into the beginning of the 1900s, one of the... industries that was really starting to take off and include much more analytics than other industries. And analytics is what we're studying because we're studying business analytics and quantitative methods.
One of the industries that was really seeing a dramatic rise in analytics was the alcohol industry. Now, there have always been very strong ties. between the alcohol industry and changes in economic, legal, and other things.
So for instance, what was one of the major complaints of the founders of the United States against England? One of the major reasons for the Revolutionary War was what? taxation, and in particular taxation for what product? Tea was one, but there was a particular product that was being produced in this country in large quantities by the people who just happened to be what we would reference as the founding fathers, alcohol. So for instance, somebody who had a very famous...
quantity of whiskey was George Washington because he was a very prolific salesperson of that his particular still so and by that I don't mean like you know don't think like somebody in the Appalachian mountains producing like small quantities of stuff no he had a giant business he made tons of money at the time And the same thing for the other founding fathers. One of the big problems they had with England is that England was trying to limit the sales of alcohol. And that was one of the ways that people could move from lower middle class to very upper middle class, if not upper class, quite quickly is through the sale of alcohol. And so England was not fond of that.
And so they were trying to then greatly tax it. So that was one of the complaints that Washington, as well as others, had. Now, you saw similar things happening in history in the late 1800s, early 1900s, of where you saw people becoming quite successful on the sales of various alcohols.
But in particular, the variety that we are going to care about for this story was the Guinness plant. So what is Guinness? How could we describe what Guinness is for those who are unaware? like a very brown beer.
So sometimes people reference it as the equivalent of drinking bread. I happen to think it's delicious. So at this particular point in time, what Guinness was doing was they were beginning to analyze their processes to try to make improvements, both in quality as well as standard.
And there was a particular person who is an analyst who is working for them, who made some discoveries in hypothesis testing about how to make statements like the following. This batch of Guinness is similar to this batch of Guinness. And that's an important statement because when you're producing a good, you want quality control so that what you're selling is always similar so that a customer doesn't say Well, this batch was really good, but that batch was terrible because you don't want that happening.
You want to be able to control the quality of your product. So that kind of statement of this batch is similar to this batch is very important to make. And so this particular analyst came up with a collection of techniques for testing things under those kinds of conditions. And he saw that it was so important, he wanted to publish his results so that people outside of just his particular plant could use them.
But the problem is, is that when you come up with that kind of innovation, you don't want your competitors to know right away. So what he did is that he asked his bosses, can I publish these results? And they said, yes, so long as it is anonymous who you are.
so that it can't be instantly traced back to our competitors, that they should be doing the same analysis on their beer. So he published his result as A Student of Statistics. That was a pseudonym that he created, A Student of Statistics. And as a result, then, this technique became referred to as the student tea distribution technique.
And I happen to think that's a very interesting story from history about where a name came from. And as we will see, this will match up quite nicely with doing these kinds of tests on whether there are differences between things. Now, one of the other techniques that we will learn is what's referred to as the ANOVA technique.
And ANOVA is a statistical method used to compare the means of three or more groups. So as I said, the t distribution kind of test and the t test is about is this group different from that group? Well that's between two things. ANOVA is a way of trying to test if I have more than two things, is there a difference between them? Okay, so let's go back up and let's review one at a time.
The concepts we've thus far talked about, and again, they're only concepts because I haven't done any examples of these things. The concepts we've talked about so far from inferential statistics. So for starters, what is inferential statistics?
Turn to your partner. What is inferential statistics? Put it into your own words.
Okay, so who could tell me what inferential statistics is? I heard good answers. The fact that I'm hard of hearing also should give confidence to anybody who's close to me that probably they could give their answer. maybe two individuals who are very close to me, who had what I thought was a very nice phrasing on it.
Now, making some kind of a prediction or claim based upon information that you've collected. Perfect. Nicely stated. That's inferential statistics.
Okay. And that also includes a nice description of what sampling is, which is where you're collecting the information. Let me come down in confidence intervals. So we talked about confidence intervals.
Turn to your partners. What's confidence intervals? Okay, so row one is exempt because individuals from row one answered the last one. So we now have three other rows in the room that could answer this question. Yeah, yeah.
So confidence is how much you believe that something in a particular range... gives you the truth. Okay. So as we said, we had our example. of where, for instance, between 5'3 and 6'1, I think that was what I said, would be the average height in the room.
And then there'd be some percentage of confidence we would have that that would be the correct estimate range. And that's really what confidence intervals are always connected to is estimate ranges. Okay, so now we come down to hypothesis testing.
So hypothesis testing is just simply the prospect of testing a claim. That's all it is. Just testing a claim, whether or not it's true. We saw zero examples of Z tests and T tests.
But what we learned is that apparently they're what you use in hypothesis testing. Cool. ANOVA.
It's just simply a way of testing for whether there are differences between more than two groups. And regression analysis, we're going to save for at least past next time. Because next time, what we're going to do is that we're going to move down to this part, which is where it runs through actually us doing some examples actually on some sets of data.
So in other words, we're going to run some hypothesis testing on actual information. So that's what we're going to do next time. So if you want to read ahead, you're very welcome to. So in terms of where we are, in terms of how you find this link, it's just simply inferential statistics.
It's the first one that's on there. We will eventually get to some of the sub links. But so off on the side, you just click on inferential statistics, and that's this page.
So. More on this next time. Have a very good one.