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So C, if we assume a few things, is a classic geometric random variable. What tells us that? Well, the giveaway is that we're gonna keep doing these independent trials where the probability of success is constant and there's a clear success. A telephone order in this case is a success. The probability is 10% of it happening. And we're gonna keep doing it until we get a success. So classic geometric random variable. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
A telephone order in this case is a success. The probability is 10% of it happening. And we're gonna keep doing it until we get a success. So classic geometric random variable. Now they ask us, find the probability, the probability that it takes fewer than five orders for Liliana to get her first telephone order of the month. So it's really the probability that C is less than five. So like always, pause this video and have a go at it. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
So classic geometric random variable. Now they ask us, find the probability, the probability that it takes fewer than five orders for Liliana to get her first telephone order of the month. So it's really the probability that C is less than five. So like always, pause this video and have a go at it. And even if you struggle with it, that's even, that's better, your brain will be more primed for the actual solution that we can go through together. All right. So I'm assuming you've had a go at it. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
So like always, pause this video and have a go at it. And even if you struggle with it, that's even, that's better, your brain will be more primed for the actual solution that we can go through together. All right. So I'm assuming you've had a go at it. So there's a couple of ways to approach it. You could say, well look, this is just gonna be the probability that C is equal to one plus the probability that C is equal to two plus the probability that C is equal to three plus the probability that C is equal to four. And we can calculate it this way. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
So I'm assuming you've had a go at it. So there's a couple of ways to approach it. You could say, well look, this is just gonna be the probability that C is equal to one plus the probability that C is equal to two plus the probability that C is equal to three plus the probability that C is equal to four. And we can calculate it this way. What is the probability that C equals one? Well, it's the probability that her very first order is a telephone order. And so we'll have 0.1. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
And we can calculate it this way. What is the probability that C equals one? Well, it's the probability that her very first order is a telephone order. And so we'll have 0.1. What's the probability that C equals two? Well, it's the probability that her first order is not a telephone order. So it's one minus 10%. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
And so we'll have 0.1. What's the probability that C equals two? Well, it's the probability that her first order is not a telephone order. So it's one minus 10%. There's a 90% chance it's not a telephone order. And that her second order is a telephone order. What about the probability C equals three? | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
So it's one minus 10%. There's a 90% chance it's not a telephone order. And that her second order is a telephone order. What about the probability C equals three? Well, her first two orders would not be telephone orders and her third order would be one. And then C equals four? Well, her first three orders would not be telephone orders and her fourth one would. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
What about the probability C equals three? Well, her first two orders would not be telephone orders and her third order would be one. And then C equals four? Well, her first three orders would not be telephone orders and her fourth one would. And we could get a calculator maybe and add all of these things up and we would actually get the answer. But you're probably wondering, well, this is kinda hairy to type into a calculator. Maybe there is an easier way to tackle this. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
Well, her first three orders would not be telephone orders and her fourth one would. And we could get a calculator maybe and add all of these things up and we would actually get the answer. But you're probably wondering, well, this is kinda hairy to type into a calculator. Maybe there is an easier way to tackle this. And indeed, there is. So think about it. The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
Maybe there is an easier way to tackle this. And indeed, there is. So think about it. The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. One minus the probability that no telephone order in first four orders. So what's this? Well, because this is just saying, what's the probability we do have an order in the first four? | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
The probability that C is less than five, that's the same thing as one minus the probability that we don't have a telephone order in the first four. One minus the probability that no telephone order in first four orders. So what's this? Well, because this is just saying, what's the probability we do have an order in the first four? So it's the same thing as one minus the probability that we don't have an order in the first four. And this is pretty straightforward to calculate. So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
Well, because this is just saying, what's the probability we do have an order in the first four? So it's the same thing as one minus the probability that we don't have an order in the first four. And this is pretty straightforward to calculate. So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. So what's the probability that we have no telephone orders in the first four orders? Well, the probability on a given order that you don't have a telephone order is 0.9. And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
So this is going to be equal to one minus, and let me do this in another color so we know what I'm referring to. So what's the probability that we have no telephone orders in the first four orders? Well, the probability on a given order that you don't have a telephone order is 0.9. And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power. So this is a lot easier to calculate. So let's do that. Let's get a calculator out. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
And then if that has to be true for the first four, well, it's gonna be 0.9 times 0.9 times 0.9 times 0.9, or 0.9 to the fourth power. So this is a lot easier to calculate. So let's do that. Let's get a calculator out. All right, so let me just take 0.9 to the fourth power is equal to, and then let me subtract that from one. So let me make that a negative, and then let me add one to it. And we get, there you go, 0.3439. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
Let's get a calculator out. All right, so let me just take 0.9 to the fourth power is equal to, and then let me subtract that from one. So let me make that a negative, and then let me add one to it. And we get, there you go, 0.3439. So this is equal to 0.3439. And we're done. That's the probability that it takes fewer than five orders for her to get her first telephone order of the month. | Cumulative geometric probability (less than a value) AP Statistics Khan Academy.mp3 |
Three of them are unfair in that they have a 45% chance of coming up tails when flipped. The rest are fair. So for the rest of them, you have a 50% chance of tails or a 50% chance of heads. You randomly choose one coin from the bag and flip it four times. What is the percent probability of getting four heads? So let's think about it. When we put our hand in the bag and we take one of the coins out, there's some probability that we get an unfair coin. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
You randomly choose one coin from the bag and flip it four times. What is the percent probability of getting four heads? So let's think about it. When we put our hand in the bag and we take one of the coins out, there's some probability that we get an unfair coin. And three of the four coins are unfair. So there's a 3 4 probability that we get an unfair coin. And then there is only one out of the four coins that's fair. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
When we put our hand in the bag and we take one of the coins out, there's some probability that we get an unfair coin. And three of the four coins are unfair. So there's a 3 4 probability that we get an unfair coin. And then there is only one out of the four coins that's fair. So there was a 1 4 probability that I get a fair coin. Now, given that I have unfair, let's remind ourselves. An unfair coin has a 45% chance of coming up tails. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
And then there is only one out of the four coins that's fair. So there was a 1 4 probability that I get a fair coin. Now, given that I have unfair, let's remind ourselves. An unfair coin has a 45% chance of coming up tails. So this means that I have a 45% chance of tails, which also means, and we have to be careful here because they're asking us about heads, if I have a 45% chance of getting tails, that means I have a 55% chance of getting heads. Whatever, I have 100% chance of getting one of these two. If it's 45% for tails, 100 minus 45 is 55 for heads. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
An unfair coin has a 45% chance of coming up tails. So this means that I have a 45% chance of tails, which also means, and we have to be careful here because they're asking us about heads, if I have a 45% chance of getting tails, that means I have a 55% chance of getting heads. Whatever, I have 100% chance of getting one of these two. If it's 45% for tails, 100 minus 45 is 55 for heads. For the fair coin, I have a 50% chance of tails and a 50% chance of heads. Fair enough. Now, I want to know, in either of these scenarios, what is the percent probability of getting four heads? | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
If it's 45% for tails, 100 minus 45 is 55 for heads. For the fair coin, I have a 50% chance of tails and a 50% chance of heads. Fair enough. Now, I want to know, in either of these scenarios, what is the percent probability of getting four heads? So if given I've got the unfair coin, the probability of getting four heads is going to be 55% for each of those flips. So the probability of getting exactly four heads is going to be 0.55 times 0.55 times 0.55 times 0.55. And so the probability of picking an unfair coin and getting four heads in a row is going to be equal to 3 4ths times all of this business over here. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
Now, I want to know, in either of these scenarios, what is the percent probability of getting four heads? So if given I've got the unfair coin, the probability of getting four heads is going to be 55% for each of those flips. So the probability of getting exactly four heads is going to be 0.55 times 0.55 times 0.55 times 0.55. And so the probability of picking an unfair coin and getting four heads in a row is going to be equal to 3 4ths times all of this business over here. So that's 3 4ths times, and this is 0.55 times itself four times. So I could write it as 0.55 to the fourth power. And we'll get the calculator out in a second to actually calculate what this is. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
And so the probability of picking an unfair coin and getting four heads in a row is going to be equal to 3 4ths times all of this business over here. So that's 3 4ths times, and this is 0.55 times itself four times. So I could write it as 0.55 to the fourth power. And we'll get the calculator out in a second to actually calculate what this is. Now, let's do the same thing for the fair coin. If I did pick a fair coin, the probability of getting heads four times in a row is going to be 0.5 times 0.5 times 0.5 times 0.5. Or the probability of getting the fair coin, which is 1 4th chance, times the probability and getting four heads in a row is going to be 1 4th times all of this. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
And we'll get the calculator out in a second to actually calculate what this is. Now, let's do the same thing for the fair coin. If I did pick a fair coin, the probability of getting heads four times in a row is going to be 0.5 times 0.5 times 0.5 times 0.5. Or the probability of getting the fair coin, which is 1 4th chance, times the probability and getting four heads in a row is going to be 1 4th times all of this. So it's going to be 1 4th times, this is just 0.5 times itself, four times. So that's 0.5 to the fourth power. So let's get the calculator out to calculate either one of these. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
Or the probability of getting the fair coin, which is 1 4th chance, times the probability and getting four heads in a row is going to be 1 4th times all of this. So it's going to be 1 4th times, this is just 0.5 times itself, four times. So that's 0.5 to the fourth power. So let's get the calculator out to calculate either one of these. So we get 3 divided by 4 times, and it knows that when I do the multiplication, it's not in the denominator here. So it's 3 4ths times, and I'll just do it in parentheses, which I don't have to do in parentheses because it knows order of operation. So 0.55 to the fourth power is equal to 0. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
So let's get the calculator out to calculate either one of these. So we get 3 divided by 4 times, and it knows that when I do the multiplication, it's not in the denominator here. So it's 3 4ths times, and I'll just do it in parentheses, which I don't have to do in parentheses because it knows order of operation. So 0.55 to the fourth power is equal to 0. So let me write it down. Let me take it off the screen so I can write it down properly. Actually, let me just do both of these calculations. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
So 0.55 to the fourth power is equal to 0. So let me write it down. Let me take it off the screen so I can write it down properly. Actually, let me just do both of these calculations. So this probability is that one right over there. And then this one down here is 1 divided by 4 times 0.5 to the fourth power. So it's equal to that right over there. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
Actually, let me just do both of these calculations. So this probability is that one right over there. And then this one down here is 1 divided by 4 times 0.5 to the fourth power. So it's equal to that right over there. So let's be clear. The probability of picking the unfair coin and then getting four heads in a row is this top number. It's like roughly 6.9% chance that you get the unfair coin and then get four heads in a row. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
So it's equal to that right over there. So let's be clear. The probability of picking the unfair coin and then getting four heads in a row is this top number. It's like roughly 6.9% chance that you get the unfair coin and then get four heads in a row. The probability that you get the fair coin and then get four heads in a row is even lower. It's only a 1.6% chance. Now, the probability of getting four heads in a row either way is going to be the sum of this and this, or the sum of that and that, which is going to be. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
It's like roughly 6.9% chance that you get the unfair coin and then get four heads in a row. The probability that you get the fair coin and then get four heads in a row is even lower. It's only a 1.6% chance. Now, the probability of getting four heads in a row either way is going to be the sum of this and this, or the sum of that and that, which is going to be. Let me keep my calculator out. So it's going to be equal to. I can just take the previous answer. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
Now, the probability of getting four heads in a row either way is going to be the sum of this and this, or the sum of that and that, which is going to be. Let me keep my calculator out. So it's going to be equal to. I can just take the previous answer. Let me just retype it so I don't confuse you. So 0.015625 plus 0.0686296875. I'm going to round it anyway, so it won't matter too much. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
I can just take the previous answer. Let me just retype it so I don't confuse you. So 0.015625 plus 0.0686296875. I'm going to round it anyway, so it won't matter too much. So if I take the sum. Let me take this off screen so I can still see it, and then let me write it. So what I got here, this one is 0.068629, and I'll round it 7. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
I'm going to round it anyway, so it won't matter too much. So if I take the sum. Let me take this off screen so I can still see it, and then let me write it. So what I got here, this one is 0.068629, and I'll round it 7. And this down here was 0.015625. And when you add these two up, because we just care about getting four heads either way, there's a probability of getting it this way with the unfair coin. This is the probability of getting it with the fair coin. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
So what I got here, this one is 0.068629, and I'll round it 7. And this down here was 0.015625. And when you add these two up, because we just care about getting four heads either way, there's a probability of getting it this way with the unfair coin. This is the probability of getting it with the fair coin. We want it either way. So let's add the two, which we already did in our calculator. So if you add that number to that number, you get 0.08425, and it keeps going. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
This is the probability of getting it with the fair coin. We want it either way. So let's add the two, which we already did in our calculator. So if you add that number to that number, you get 0.08425, and it keeps going. But I'm just going to round it. So this is the same thing as this is equal to 8.425% if I want to round it a little bit more, 8.43% chance of getting four heads in a row. And once again, that's a slightly higher number than if all of the coins were fair, because there's a 3 4th chance that I get a coin that has a better than even chance of getting heads. | Dependent probability example 2 Probability and Statistics Khan Academy.mp3 |
The company gathered the following data about consumers' preference of soda. So they have year by year, percentage of respondents who preferred Yummy Cola, percentage of respondents who preferred Thrill Cola, and then these are people who had no preference. So in 2006, 80% liked Yummy, only 12% liked Thrill, and 8% didn't like either one or didn't have any preference. And so actually just from here you see that many, many more people liked Yummy Cola than Thrill Cola, actually every year over here. So Thrill Cola definitely has something, they have an uphill battle. But then they said the advertising company created the following two graphs to promote Thrill Soda. And so let's see what's happening over here. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And so actually just from here you see that many, many more people liked Yummy Cola than Thrill Cola, actually every year over here. So Thrill Cola definitely has something, they have an uphill battle. But then they said the advertising company created the following two graphs to promote Thrill Soda. And so let's see what's happening over here. So let's think about whether this is misleading or not. So if we look at this graph over here, in 2006, sure enough, 80% liked Yummy Cola, then 2007, 76%, then it keeps going, then 77%, then 73%, then 73, then 68. So this is actually accurate data. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
And so let's see what's happening over here. So let's think about whether this is misleading or not. So if we look at this graph over here, in 2006, sure enough, 80% liked Yummy Cola, then 2007, 76%, then it keeps going, then 77%, then 73%, then 73, then 68. So this is actually accurate data. It actually represents the data that's given right here. I'll do it in the same, it actually represents this data very faithfully. Then right over here, if we look at this chart, percentage of people who prefer Thrill Soda. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So this is actually accurate data. It actually represents the data that's given right here. I'll do it in the same, it actually represents this data very faithfully. Then right over here, if we look at this chart, percentage of people who prefer Thrill Soda. So over here in 2006, 12% preferred Thrill Soda, 2006, 12%, 2007, 19%, 2008, 19%, then we go up to 20, 21, and 25. So the graphs are actually accurate. They're not lying. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
Then right over here, if we look at this chart, percentage of people who prefer Thrill Soda. So over here in 2006, 12% preferred Thrill Soda, 2006, 12%, 2007, 19%, 2008, 19%, then we go up to 20, 21, and 25. So the graphs are actually accurate. They're not lying. These are actually the data points of the percentage who prefer Thrill Soda. Now what's misleading is if someone were to just look at these two graphs without actually looking at the scales over here, they'll see two things. They'll say, oh look, you see a declining trend, and that's what line graphs are good for, for seeing trends. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
They're not lying. These are actually the data points of the percentage who prefer Thrill Soda. Now what's misleading is if someone were to just look at these two graphs without actually looking at the scales over here, they'll see two things. They'll say, oh look, you see a declining trend, and that's what line graphs are good for, for seeing trends. They say, look, I see a declining trend in the number of, in the percentage of people who prefer Yummy Cola, and I see this increasing trend in the percentage of people who prefer Thrill Cola. And that's true. You have a declining trend here, and you have an increasing trend here. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
They'll say, oh look, you see a declining trend, and that's what line graphs are good for, for seeing trends. They say, look, I see a declining trend in the number of, in the percentage of people who prefer Yummy Cola, and I see this increasing trend in the percentage of people who prefer Thrill Cola. And that's true. You have a declining trend here, and you have an increasing trend here. But what's misleading here is the way that they've plotted the scales. These scales are not the same. So when you look at this, you say not only is there an increasing trend of people who prefer Thrill Soda, but the way they set up the scale, it looks like the trend is above. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
You have a declining trend here, and you have an increasing trend here. But what's misleading here is the way that they've plotted the scales. These scales are not the same. So when you look at this, you say not only is there an increasing trend of people who prefer Thrill Soda, but the way they set up the scale, it looks like the trend is above. It looks like if you look, the human brain is tempted to compare these, and they say, look, not only is this an upward trend, but it's above this trend right over here. Even in 2006, this data point looks higher than these data points right over here. But the reality is that it's only because the scale is distorted. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So when you look at this, you say not only is there an increasing trend of people who prefer Thrill Soda, but the way they set up the scale, it looks like the trend is above. It looks like if you look, the human brain is tempted to compare these, and they say, look, not only is this an upward trend, but it's above this trend right over here. Even in 2006, this data point looks higher than these data points right over here. But the reality is that it's only because the scale is distorted. And this is the oldest trick in the book when plotting line graphs. It all depends on the scale. So this just looks good because they used this scale that went from 0 to 30 as opposed to 0, 100. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
But the reality is that it's only because the scale is distorted. And this is the oldest trick in the book when plotting line graphs. It all depends on the scale. So this just looks good because they used this scale that went from 0 to 30 as opposed to 0, 100. The better thing to do, or the more genuine thing to do, or the more honest thing to do would have actually been to plot them on the same graph. Although if they did that, that wouldn't have painted a very good picture for Thrill Soda. So if we plotted it on the same graph, Thrill Soda, let's try that out. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So this just looks good because they used this scale that went from 0 to 30 as opposed to 0, 100. The better thing to do, or the more genuine thing to do, or the more honest thing to do would have actually been to plot them on the same graph. Although if they did that, that wouldn't have painted a very good picture for Thrill Soda. So if we plotted it on the same graph, Thrill Soda, let's try that out. So in 2006, 12, and actually this is even worse. You actually wouldn't even be able to plot Thrill Soda on this graph because they started this graph right over here at 50%. They didn't even start it at 0%. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So if we plotted it on the same graph, Thrill Soda, let's try that out. So in 2006, 12, and actually this is even worse. You actually wouldn't even be able to plot Thrill Soda on this graph because they started this graph right over here at 50%. They didn't even start it at 0%. So you actually would not even be able to plot Thrill Soda on this graph. If you did, you would have to extend this graph all the way down. So you would have to extend this graph all the way down to, you know, so this would have to be 40, this would be 30, this would be 20%, this would be 10%, and then down all the way over here would be 0%. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
They didn't even start it at 0%. So you actually would not even be able to plot Thrill Soda on this graph. If you did, you would have to extend this graph all the way down. So you would have to extend this graph all the way down to, you know, so this would have to be 40, this would be 30, this would be 20%, this would be 10%, and then down all the way over here would be 0%. And then the Thrill Soda graph would be all the way down here. So it was like 12% and it goes all the way up to like 25%. So the Thrill Soda, so it would have looked something like this. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So you would have to extend this graph all the way down to, you know, so this would have to be 40, this would be 30, this would be 20%, this would be 10%, and then down all the way over here would be 0%. And then the Thrill Soda graph would be all the way down here. So it was like 12% and it goes all the way up to like 25%. So the Thrill Soda, so it would have looked something like this. The graph would have looked something like this, which is nowhere near, if you plotted these on the same scale, on the same graph, then it would have still been pretty obvious that a lot more people, even though the trend is downward, a lot more people prefer Yummy Cola to Thrill Cola. So there's two very disingenuous things going on over here. One is the actual scale. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So the Thrill Soda, so it would have looked something like this. The graph would have looked something like this, which is nowhere near, if you plotted these on the same scale, on the same graph, then it would have still been pretty obvious that a lot more people, even though the trend is downward, a lot more people prefer Yummy Cola to Thrill Cola. So there's two very disingenuous things going on over here. One is the actual scale. For this amount of distance on this scale, they represent 10%. So whatever the gain is, it looks like it's a huge gain. But over here, that same amount, they're actually representing a larger amount. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
One is the actual scale. For this amount of distance on this scale, they represent 10%. So whatever the gain is, it looks like it's a huge gain. But over here, that same amount, they're actually representing a larger amount. They're representing closer to 15% or 16%. And then the main thing is they started the scale here at 50%. So they're not showing how many people really prefer, how large 80% or even 70% really is. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
But over here, that same amount, they're actually representing a larger amount. They're representing closer to 15% or 16%. And then the main thing is they started the scale here at 50%. So they're not showing how many people really prefer, how large 80% or even 70% really is. And over here they start at 0% and they just have a larger scale. So it makes it look like out the gate a lot of people prefer, or a comparable amount of people prefer Thrill, and that the trend is up. But the reality is still way more people prefer Yummy Cola. | Misleading line graphs Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
In this video, I want to give you an example of what it means to fit data to a line. Instead of doing my traditional video using my little pen tablet, I'm going to do it straight on Excel so you could see how to do this for yourself if you have Excel or some other type of spreadsheet program. And we're not going to go into the math of it. I really just want you to get the conceptual understanding of what it means to fit data with line or to do a linear regression. So here, let's just read the problem. The following table shows the median California income. Remember, median is the middle California income from 1995 to 2002, as reported by the US Census Bureau. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
I really just want you to get the conceptual understanding of what it means to fit data with line or to do a linear regression. So here, let's just read the problem. The following table shows the median California income. Remember, median is the middle California income from 1995 to 2002, as reported by the US Census Bureau. Draw a scatter plot and find the equation. What would you expect the median annual income of a California family to be in the year 2010? What are the meanings of the slope and the y-intercept of this problem? | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
Remember, median is the middle California income from 1995 to 2002, as reported by the US Census Bureau. Draw a scatter plot and find the equation. What would you expect the median annual income of a California family to be in the year 2010? What are the meanings of the slope and the y-intercept of this problem? So the first thing you'd want to do, I just copied and pasted this image. We have to get the data in a form that the spreadsheet can understand it. So let's make some tables here. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
What are the meanings of the slope and the y-intercept of this problem? So the first thing you'd want to do, I just copied and pasted this image. We have to get the data in a form that the spreadsheet can understand it. So let's make some tables here. Let's say years since 1995. Let's make that one column. Let me make this a little bit wider. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So let's make some tables here. Let's say years since 1995. Let's make that one column. Let me make this a little bit wider. And then let me put median income. This is the median income in California for a family. So we start off with one year or zero years since 1995. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
Let me make this a little bit wider. And then let me put median income. This is the median income in California for a family. So we start off with one year or zero years since 1995. Zero, one, two, three, four. And actually, if you want, it'll figure out the trend. If you just keep going down, it'll figure out that you're just incrementing by one. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So we start off with one year or zero years since 1995. Zero, one, two, three, four. And actually, if you want, it'll figure out the trend. If you just keep going down, it'll figure out that you're just incrementing by one. And then the income, I'll just copy in these numbers right there. So that's $53,807, $55,217, $55,209, $55,415, $63,100, $63,206, $63,761, and then we have $65,766. So I don't need these over here. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
If you just keep going down, it'll figure out that you're just incrementing by one. And then the income, I'll just copy in these numbers right there. So that's $53,807, $55,217, $55,209, $55,415, $63,100, $63,206, $63,761, and then we have $65,766. So I don't need these over here. So I'm going to get rid of them. I can clear them. So let me make sure I have enough entries. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So I don't need these over here. So I'm going to get rid of them. I can clear them. So let me make sure I have enough entries. This is one, two, three, four, five, six, seven, eight. And I have one, two, three, four, five, six, seven, eight entries, and I want to make sure I got my data right. $53,807, $55,217, $55,209, $45,415, $100, $206, $761, $766. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So let me make sure I have enough entries. This is one, two, three, four, five, six, seven, eight. And I have one, two, three, four, five, six, seven, eight entries, and I want to make sure I got my data right. $53,807, $55,217, $55,209, $45,415, $100, $206, $761, $766. OK, there we go. Now, you're going to find that in Excel this is incredibly easy, if you know what to click on, to one, plot this data, create a scatter plot, and then even better, create a regression of that data. So all you have to do is you select the data, and then you go to Insert, and I'm going to insert a scatter plot. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
$53,807, $55,217, $55,209, $45,415, $100, $206, $761, $766. OK, there we go. Now, you're going to find that in Excel this is incredibly easy, if you know what to click on, to one, plot this data, create a scatter plot, and then even better, create a regression of that data. So all you have to do is you select the data, and then you go to Insert, and I'm going to insert a scatter plot. And then you can pick the different types of scatter plots. I just want to plot the data. And there you go. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So all you have to do is you select the data, and then you go to Insert, and I'm going to insert a scatter plot. And then you can pick the different types of scatter plots. I just want to plot the data. And there you go. It plotted the data for me. So there you go. If you go by, this is the actual income, and this is by year since 1995. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
And there you go. It plotted the data for me. So there you go. If you go by, this is the actual income, and this is by year since 1995. So this is 1995. It was $53,807. In 1996, it's $55,217. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
If you go by, this is the actual income, and this is by year since 1995. So this is 1995. It was $53,807. In 1996, it's $55,217. So it plotted all the data. Now what I want to do is fit a line. So this isn't exactly a line, but let's see, if we assume that a line can model this data well, I'm going to get Excel to fit a line for me. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
In 1996, it's $55,217. So it plotted all the data. Now what I want to do is fit a line. So this isn't exactly a line, but let's see, if we assume that a line can model this data well, I'm going to get Excel to fit a line for me. So what I can do is I have all of these options up here for different ways to fit a line, all of these different options, and I'm going to pick this one here. You might not be able to see it. It looks like it has a line between dots, and it also has fx, which tells me it's going to tell me the equation of the line. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So this isn't exactly a line, but let's see, if we assume that a line can model this data well, I'm going to get Excel to fit a line for me. So what I can do is I have all of these options up here for different ways to fit a line, all of these different options, and I'm going to pick this one here. You might not be able to see it. It looks like it has a line between dots, and it also has fx, which tells me it's going to tell me the equation of the line. So if I click on that, there you go. It not only fit, it re-plotted that same data on a different graph. Let me make it a little bit bigger. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
It looks like it has a line between dots, and it also has fx, which tells me it's going to tell me the equation of the line. So if I click on that, there you go. It not only fit, it re-plotted that same data on a different graph. Let me make it a little bit bigger. We can cover up the data now just because I think we know what's going on. So let me cover it up right like that. So not only did it plot the various data points, it actually fit a line to that data, and it gave me the equation of that line. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
Let me make it a little bit bigger. We can cover up the data now just because I think we know what's going on. So let me cover it up right like that. So not only did it plot the various data points, it actually fit a line to that data, and it gave me the equation of that line. It says the equation of this line is y. Let me see if I can make this a little bit bigger. I'll move it out of the way so you can read it at least. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So not only did it plot the various data points, it actually fit a line to that data, and it gave me the equation of that line. It says the equation of this line is y. Let me see if I can make this a little bit bigger. I'll move it out of the way so you can read it at least. So it tells me right here that the equation for this line is y is equal to 1,882.3x plus 52,847. So if you remember what we know about slope and y-intercept, the y-intercept is 52,847, which is if you use this line as your measure, where this line intersects at year 0, or in 1995. So if you use this line as a model, in 1995 the line would say that you're going to make 52,847. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
I'll move it out of the way so you can read it at least. So it tells me right here that the equation for this line is y is equal to 1,882.3x plus 52,847. So if you remember what we know about slope and y-intercept, the y-intercept is 52,847, which is if you use this line as your measure, where this line intersects at year 0, or in 1995. So if you use this line as a model, in 1995 the line would say that you're going to make 52,847. The actual data was a little bit off of that. It was a little bit higher, 53,807. So it was a little bit higher, but we're trying to get a line that gets as close as possible to all of this data. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So if you use this line as a model, in 1995 the line would say that you're going to make 52,847. The actual data was a little bit off of that. It was a little bit higher, 53,807. So it was a little bit higher, but we're trying to get a line that gets as close as possible to all of this data. It's actually trying to minimize the distance, the square of the distance, between each of these points in the line. And we won't go into the math there. But it gave us this nice equation. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So it was a little bit higher, but we're trying to get a line that gets as close as possible to all of this data. It's actually trying to minimize the distance, the square of the distance, between each of these points in the line. And we won't go into the math there. But it gave us this nice equation. Now we can use this nice equation to predict things. If we say that this is a good model for the data, let me bring this down a little bit, let's try to answer our question. So we drew a scatter plot, really Excel did it for us. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
But it gave us this nice equation. Now we can use this nice equation to predict things. If we say that this is a good model for the data, let me bring this down a little bit, let's try to answer our question. So we drew a scatter plot, really Excel did it for us. We found the equation right there. They say, what would you expect the median annual income of a California family to be in the year 2010? So here we can just use the equation they gave us. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So we drew a scatter plot, really Excel did it for us. We found the equation right there. They say, what would you expect the median annual income of a California family to be in the year 2010? So here we can just use the equation they gave us. This right here was 2002. So I could write down the year. This was the year of 2002. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So here we can just use the equation they gave us. This right here was 2002. So I could write down the year. This was the year of 2002. So the year 2010 is 8 more years. And let me make a little column here. So this is the year, 1995, 1996. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
This was the year of 2002. So the year 2010 is 8 more years. And let me make a little column here. So this is the year, 1995, 1996. And then Excel will be able to figure out, if I select those and I go to this little bottom right square, and I scroll down, Excel will actually figure out that I want to increment by 1 year every time. And if I say years since 1995, once again I can just continue this trend right here. So 2010 would be 15 years. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So this is the year, 1995, 1996. And then Excel will be able to figure out, if I select those and I go to this little bottom right square, and I scroll down, Excel will actually figure out that I want to increment by 1 year every time. And if I say years since 1995, once again I can just continue this trend right here. So 2010 would be 15 years. And so we can just apply this equation. We could say it's going to be equal to, according to this line. I'm just going to type it in. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So 2010 would be 15 years. And so we can just apply this equation. We could say it's going to be equal to, according to this line. I'm just going to type it in. Hopefully you can read what I'm saying. 1,882.3 times x. x here is the year since 1995. So times, I could just select this cell. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
I'm just going to type it in. Hopefully you can read what I'm saying. 1,882.3 times x. x here is the year since 1995. So times, I could just select this cell. Or I could type in the number 15. That means times this cell, times 15. And then plus 52,847. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So times, I could just select this cell. Or I could type in the number 15. That means times this cell, times 15. And then plus 52,847. Plus that right there. Click Enter. And it predicts $81,081.50. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
And then plus 52,847. Plus that right there. Click Enter. And it predicts $81,081.50. So if you just continue this line for another 8 or so years, it predicts that the median income in California for a family will be $81,000. Anyway, hopefully you found that interesting. This is, spreadsheets are very useful tools for manipulating data. | Fitting a line to data Regression Probability and Statistics Khan Academy.mp3 |
So I have some data here in a spreadsheet. You could use Microsoft Excel or you could use Google Spreadsheets. And we're gonna use the spreadsheet to quickly calculate some parameters. Let's say this is the population. Let's say this is, we're looking at a population of students and we wanna calculate some parameters, and this is their ages, and we wanna calculate some parameters on that. And so, first I'm gonna calculate it using the spreadsheet. And then we're gonna think about how those parameters change as we do things to the data. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Let's say this is the population. Let's say this is, we're looking at a population of students and we wanna calculate some parameters, and this is their ages, and we wanna calculate some parameters on that. And so, first I'm gonna calculate it using the spreadsheet. And then we're gonna think about how those parameters change as we do things to the data. If we were to shift the data up or down, or if we were to multiply all the points by some value, what does that do to the actual parameters? So the first parameter I'm gonna calculate is the mean. Then I'm gonna calculate the standard deviation. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
And then we're gonna think about how those parameters change as we do things to the data. If we were to shift the data up or down, or if we were to multiply all the points by some value, what does that do to the actual parameters? So the first parameter I'm gonna calculate is the mean. Then I'm gonna calculate the standard deviation. Then I wanna calculate the median. And then I wanna calculate, let's say the interquartile range. Inter, I'll call it IQR. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Then I'm gonna calculate the standard deviation. Then I wanna calculate the median. And then I wanna calculate, let's say the interquartile range. Inter, I'll call it IQR. So let's do this. Let's first look at the measures of central tendency. So the mean, the function on most spreadsheets is the average function. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Inter, I'll call it IQR. So let's do this. Let's first look at the measures of central tendency. So the mean, the function on most spreadsheets is the average function. And then I could use my mouse and select all of these, or I could press shift with my arrow button and select all of those. Okay, that's the mean of that data. Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So the mean, the function on most spreadsheets is the average function. And then I could use my mouse and select all of these, or I could press shift with my arrow button and select all of those. Okay, that's the mean of that data. Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it. So if I took all the data and if I were to add five to it. So an easy way to do that in a spreadsheet is you select that, you add five, and then I can scroll down. And notice, for every data point I had before, I now have five more than that. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Now let's think about what happens if I take all of that data and if I were to add a fixed amount to it. So if I took all the data and if I were to add five to it. So an easy way to do that in a spreadsheet is you select that, you add five, and then I can scroll down. And notice, for every data point I had before, I now have five more than that. So this is my new data set, which I'm calling data plus five. And let's see what the mean of that is. So the mean of that, notice, is exactly five more. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
And notice, for every data point I had before, I now have five more than that. So this is my new data set, which I'm calling data plus five. And let's see what the mean of that is. So the mean of that, notice, is exactly five more. And the same would have been true if I added or subtracted any number. The mean would change by the amount that I add or subtract. And so, and that shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by the numbers you have. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So the mean of that, notice, is exactly five more. And the same would have been true if I added or subtracted any number. The mean would change by the amount that I add or subtract. And so, and that shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by the numbers you have. And so, if all the numbers are five more, you're gonna add five, in this case, how many numbers are there? One, two, three, four, five, six, seven, eight, nine, 10, 11, 12. You're gonna add 12 more fives and then you're gonna divide by 12, and so it makes sense that your mean goes up by five. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
And so, and that shouldn't surprise you, because when you're calculating the mean, you're adding all the numbers up and you're dividing by the numbers you have. And so, if all the numbers are five more, you're gonna add five, in this case, how many numbers are there? One, two, three, four, five, six, seven, eight, nine, 10, 11, 12. You're gonna add 12 more fives and then you're gonna divide by 12, and so it makes sense that your mean goes up by five. Let's think about how the mean changes if you multiply. So if you take your data and if I were to multiply it times five, what happens? So this equals this times five. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
You're gonna add 12 more fives and then you're gonna divide by 12, and so it makes sense that your mean goes up by five. Let's think about how the mean changes if you multiply. So if you take your data and if I were to multiply it times five, what happens? So this equals this times five. So now, all the data points are five times more. Now what happens to my mean? Notice, my mean is now five times as much. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So this equals this times five. So now, all the data points are five times more. Now what happens to my mean? Notice, my mean is now five times as much. So the measures of central tendency, if I add or subtract, well, I'm gonna add or subtract the mean by that amount. And if I scale it up by five or if I scaled it down by five, well, my mean would scale up or down by that same amount. And if you numerically looked at how you calculate a mean, it would make sense that this is happening mathematically. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Notice, my mean is now five times as much. So the measures of central tendency, if I add or subtract, well, I'm gonna add or subtract the mean by that amount. And if I scale it up by five or if I scaled it down by five, well, my mean would scale up or down by that same amount. And if you numerically looked at how you calculate a mean, it would make sense that this is happening mathematically. Now let's look at the other measure, the other typical measure of central tendency, and that is the median, to see if that has the same properties. So let's calculate the median here. So once again, you wanna order these numbers and just find the middle number, which isn't too hard, but a computer can do it awfully fast. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
And if you numerically looked at how you calculate a mean, it would make sense that this is happening mathematically. Now let's look at the other measure, the other typical measure of central tendency, and that is the median, to see if that has the same properties. So let's calculate the median here. So once again, you wanna order these numbers and just find the middle number, which isn't too hard, but a computer can do it awfully fast. So that's the median for that data set. What do you think the median's gonna be if you take all of the data plus five? Well, the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the middle is gonna be five more. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
So once again, you wanna order these numbers and just find the middle number, which isn't too hard, but a computer can do it awfully fast. So that's the median for that data set. What do you think the median's gonna be if you take all of the data plus five? Well, the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the middle is gonna be five more. So this should be 10.5. And yes, it is indeed 10.5. And what would happen if you multiply everything by five? | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
Well, the middle number, if you ordered all of these numbers and made them all five more, the orders, you could think of it as being the same order, but now the one in the middle is gonna be five more. So this should be 10.5. And yes, it is indeed 10.5. And what would happen if you multiply everything by five? Well, once again, you still have the same ordering, and so it should just multiply that by five. Yep, the middle number is now gonna be five times larger. So both of these measures of central tendency, if you shift all the data points or if you scale them up, you're going to similarly shift or scale up these measures of central tendency. | How parameters change as data is shifted and scaled AP Statistics Khan Academy.mp3 |
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