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These actually have fairly technical definitions when you get further in statistics, but a, I guess, easier to process version of them are when you have a left tail, you tend to be, when you are left-tailed, you also tend to be skewed to the left. And when you are right-tailed, you tend to be skewed to the right. Anoth... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
That might not make any sense to you. You might just want to go after the tail. If you're left-tailed, you're probably left-skewed. If you're right-tailed, you're probably right-skewed. So let's keep going. Let's see if we can see, let's actually see another example. So this is interesting. | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
If you're right-tailed, you're probably right-skewed. So let's keep going. Let's see if we can see, let's actually see another example. So this is interesting. This is not, we're not given a histogram here. We're not given a bar graph. We're given a box and whiskers plot, which is really just telling us the different q... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
So this is interesting. This is not, we're not given a histogram here. We're not given a bar graph. We're given a box and whiskers plot, which is really just telling us the different quartiles. So just to remind ourselves, this tells us the minimum of our data set, the bottom of our range. So the minimum value in our d... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
We're given a box and whiskers plot, which is really just telling us the different quartiles. So just to remind ourselves, this tells us the minimum of our data set, the bottom of our range. So the minimum value in our data set, we have at least 111. And then the maximum value of our data set, we have at least 125. Now... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
And then the maximum value of our data set, we have at least 125. Now this line right over here is the median. The middle number is 21. And then the box defines the middle 50% of our numbers. So it's kind of the meat of our distribution. So if we were to try to visualize what this would look like as maybe a histogram, ... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
And then the box defines the middle 50% of our numbers. So it's kind of the meat of our distribution. So if we were to try to visualize what this would look like as maybe a histogram, and we don't know for sure, because we might have a whole bunch of 11s, not so much that it skews this, but we could have more than one.... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
But a distribution that this could match up with is something that looks like having a tail down here, and then you kind of bump up here. This is the meat of the distribution. It kind of looks something like that. And I can't draw, because I'm doing this on the exercises right now. But for something like that, well, so... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
And I can't draw, because I'm doing this on the exercises right now. But for something like that, well, something like that would have a tail to the left. Would have a tail to the left. It has, its range goes, you know, fairly low to the left, but it might not have a lot of values there. If it had more values on the le... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
It has, its range goes, you know, fairly low to the left, but it might not have a lot of values there. If it had more values on the left side, this box would have been shifted over, because a larger percentage would have fit, would have been on the left, so to speak. And so this one, I feel pretty good about saying thi... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
It's definitely not symmetrical. If it was symmetrical, the median would be pretty close to the center. The box would be pretty centered. And it's not skewed to the right. If it was skewed to the right, you would have a tail to the right. You would have, this whisker would likely be much, much, much longer. And we're d... | Thinking about shapes of distributions Data and statistics 6th grade Khan Academy.mp3 |
There's a parameter here. Let's say it's the population mean. We do not know what this is, so we take a sample. Here we're gonna take a sample of 15. So n is equal to 15, and from that sample, we can calculate a sample mean. But we also wanna construct a 98% confidence interval about that sample mean. So we're gonna go... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
Here we're gonna take a sample of 15. So n is equal to 15, and from that sample, we can calculate a sample mean. But we also wanna construct a 98% confidence interval about that sample mean. So we're gonna go take that sample mean, and we're gonna go plus or minus some margin of error. Now, in other videos, we have tal... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
So we're gonna go take that sample mean, and we're gonna go plus or minus some margin of error. Now, in other videos, we have talked about that we wanna use the t distribution here because we don't want to underestimate the margin of error. So it's going to be t star times the sample standard deviation divided by the s... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
What is the t star that we should use in this situation? And so we're about to look at a, I guess we call it a t table instead of a z table, but the key thing to realize is there's one extra variable to take into consideration when we're looking up the appropriate critical value on a t table, and that's this notion of ... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
And I'm not gonna go in-depth on degrees of freedom. It's actually a pretty deep concept, but it's this idea that you actually have a different t distribution depending on the different sample sizes, depending on the degrees of freedom. And your degree of freedom is going to be your sample size minus one. So in this si... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
So in this situation, our degree of freedom is going to be 15 minus one. So in this situation, our degree of freedom is going to be equal to 14. And this isn't the first time that we have seen this. We talked a little bit about degrees of freedom when we first talked about sample standard deviations and how to have an ... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
We talked a little bit about degrees of freedom when we first talked about sample standard deviations and how to have an unbiased estimate for the population standard deviation. And in future videos, we'll go into more advanced conversations about degrees of freedom. But for the purposes of this example, you need to kn... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
When we're thinking about a confidence interval for your mean. So now let's look at the t table. So we want a 98% confidence interval. And we want a degree of freedom of 14. So let's get our t table out. And I actually copy and pasted this bottom part, moved it up so that you could see the whole thing here. And what's ... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
And we want a degree of freedom of 14. So let's get our t table out. And I actually copy and pasted this bottom part, moved it up so that you could see the whole thing here. And what's useful about this t table is they actually give our confidence levels right over here. So if you want a confidence level of 98%, you're... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
And what's useful about this t table is they actually give our confidence levels right over here. So if you want a confidence level of 98%, you're going to look at this column. You're going to look at this column right over here. Another way of thinking about a confidence level of 98%, if you have a confidence level of... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
Another way of thinking about a confidence level of 98%, if you have a confidence level of 98%, that means you're leaving 1% unfilled in at either end of the tail. And so if you're looking at your t distribution, everything up to and including that top 1%, you would look for a tail probability of 0.01, which is, you ca... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
But either way, we're in this column right over here. We have a confidence level of 98%. And remember, our degrees of freedom, our degree of freedom here is, we have 14 degrees of freedom. And so we'll look at this row right over here. And so there you have it. This is our critical t value, 2.624. And so let's just go ... | Example finding critical t value Confidence intervals AP Statistics Khan Academy.mp3 |
Consider the following story. Bob is in a room and he has two coins. One fair coin and one double-sided coin. He picks one at random, flips it, and shouts the result. Heads! Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree. | Conditional Probability.mp3 |
He picks one at random, flips it, and shouts the result. Heads! Now what is the probability that he flipped the fair coin? To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The... | Conditional Probability.mp3 |
To answer this question, we need only rewind and grow a tree. The first event, he picks one of two coins. So our tree grows two branches, leading to two equally likely outcomes, fair or unfair. The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely ... | Conditional Probability.mp3 |
The next event, he flips the coin. We grow again. If he had the fair coin, we know this flip can result in two equally likely outcomes, heads and tails. While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final ste... | Conditional Probability.mp3 |
While the unfair coin results in two outcomes, both heads. Our tree is finished and we see it has four leaves, representing four equally likely outcomes. The final step, new evidence. He says, Heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur... | Conditional Probability.mp3 |
He says, Heads! Whenever we gain evidence, we must trim our tree. We cut any branch leading to tails because we know tails did not occur. And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What h... | Conditional Probability.mp3 |
And that is it. So the probability that he chose the fair coin is the one fair outcome leading to heads, divided by the three possible outcomes leading to heads, or one third. What happens if he flips again and reports, Heads! Remember, after each event, our tree grows. The fair coin leaves result in two equally likely... | Conditional Probability.mp3 |
Remember, after each event, our tree grows. The fair coin leaves result in two equally likely outcomes, heads and tails. The unfair coin leaves result in two equally likely outcomes, heads and heads. After we hear the second, heads, we cut any branches leading to tails. Therefore, the probability the coin is fair after... | Conditional Probability.mp3 |
After we hear the second, heads, we cut any branches leading to tails. Therefore, the probability the coin is fair after two heads in a row, is the one fair outcome leading to heads, divided by all possible outcome leading to heads, or one fifth. Notice our confidence in the fair coin is dropping as more heads occur, t... | Conditional Probability.mp3 |
No matter how many flips occur, we can never be 100% certain the coin is unfair. In fact, all conditional probability questions can be solved by growing trees. Let's do one more to be sure. Bob has three coins. Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third. He c... | Conditional Probability.mp3 |
Bob has three coins. Two are fair. One is biased, which is weighted to land heads two thirds of the time and tails one third. He chooses a coin at random and flips it. Heads! Now, what is the probability he chose the biased coin? Let's rewind and build a tree. | Conditional Probability.mp3 |
He chooses a coin at random and flips it. Heads! Now, what is the probability he chose the biased coin? Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes. Fair coin, fair coin, and unfair coin. The next event, the coin is flipped. | Conditional Probability.mp3 |
Let's rewind and build a tree. The first event, choosing the coin, can lead to three equally likely outcomes. Fair coin, fair coin, and unfair coin. The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves. Two represen... | Conditional Probability.mp3 |
The next event, the coin is flipped. Each fair coin leads to two equally likely leaves, heads and tails. The biased coin leads to three equally likely leaves. Two representing heads and one representing tails. Now the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of ea... | Conditional Probability.mp3 |
Two representing heads and one representing tails. Now the trick is to always make sure our tree is balanced, meaning an equal amount of leaves growing out of each branch. To do this, we simply scale up the number of branches to the least common multiple. For two and three, this is six. And finally, we label our leaves... | Conditional Probability.mp3 |
For two and three, this is six. And finally, we label our leaves. The fair coin now splits into six equally likely leaves, three heads and three tails. For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, Heads! this new evidence allows us to trim all branche... | Conditional Probability.mp3 |
For the biased coin, we now have two tail leaves and four head leaves, and that is it. When Bob shows the result, Heads! this new evidence allows us to trim all branches, leading to tails, since tails did not occur. So the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the... | Conditional Probability.mp3 |
So the probability that he chose the biased coin, given heads occur? Well, four leaves can come from the biased coin, divided by all possible leaves. Four divided by ten, or 40%. When in doubt, it's always possible to answer conditional probability questions by Bayes' theorem. It tells us the probability of event A, gi... | Conditional Probability.mp3 |
What I wanna talk about in this video, it's really about building even more intuition, is get a gut feeling for why this independence is important for making this claim. And to get that intuition, let's look at two random variables that are definitely random variables, but that are definitely not independent. So let's ... | Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3 |
And let's say that Y is equal to the number of hours that same person, person was awake yesterday. And appreciate why these are not independent random variables. One of them is going to completely determine the other. If I slept eight hours yesterday, then I would have been awake for 16 hours. If I slept for 16 hours, ... | Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3 |
If I slept eight hours yesterday, then I would have been awake for 16 hours. If I slept for 16 hours, then I would have been awake for eight hours. We know that X plus Y, even though they're random variables, and there could be variation in X, and there could be variation in Y, but for any given person, remember, these... | Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3 |
So these are not independent, not independent. If you're given one of the variables, it would completely determine what the other variable is. The probability of getting a certain value for one variable is going to be very different given what value you got for the other variable. So they're not independent at all. So ... | Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3 |
So they're not independent at all. So in this situation, if someone said, let's just say, for the sake of argument, that the variance of X, the variance of X, is equal to, I don't know, let's say it's equal to four, and the units for variance would be squared hours, so four hours squared. We could say that the standard... | Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3 |
And let's say that the variance, let's say the standard deviation of Y is also equal to two hours. And let's say that the variance of Y, variance of Y, well, it would be the square of the standard deviation, so it would be four hours, four hours squared would be our units. So if we just tried to blindly say, oh, I'm ju... | Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3 |
So it would be four plus four. So is it equal to eight hours squared? Well, that doesn't make any sense, because we know that a random variable that is equal to X plus Y, that this is always going to be 24 hours. In fact, it's not going to have any variation. X plus Y is always gonna be 24 hours. So for these two rando... | Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3 |
In fact, it's not going to have any variation. X plus Y is always gonna be 24 hours. So for these two random variables, because they are so connected, they are not independent at all, this is actually going to be zero. There is zero variance here. X plus Y is always going to be 24, at least on Earth, where we have a 24... | Intuition for why independence matters for variance of sum AP Statistics Khan Academy.mp3 |
The two-way frequency table below shows data on type of vehicle driven, so this is type of vehicle driven, and whether there was an accident in the last year. So whether there was an accident in the last year for customers of all American auto insurance. Complete the following two-way table of column relative frequenci... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
So let's see what they're saying. They're saying, let's see, of the accidents within the last year, 28 were the people were driving an SUV, a sport utility vehicle, and 35 were in a sports car. Of the no accidents in the last year, 97 were an SUV, and 104 were a sports car. Or another way you could think of it, of the ... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
Or another way you could think of it, of the sport utility vehicles that were driven, and the total, let's see, it's 28 plus 97, which is going to be 125, of that 125, 28 had an accident within the last year, and 97 did not have an accident in the last year. Similarly, you could say of the 139 sports cars, 35 had an ac... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
So the way we could think about it, one right over here, this represents all of the sport utility vehicles. So one way you could think about it, that represents the whole universe of the sport utility vehicles, or at least the universe that this table shows. So that's really representative of the 28 plus 97. And so in ... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
And so in each of these, we want to put the relative frequency. So this right over here is going to be 28, 28 divided by the total. Notice over here it was 28, but we want this number to be the fraction of the total. Well, the fraction of the total is going to be 28 over 97 plus 28, which of course is going to be 125. ... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
Well, the fraction of the total is going to be 28 over 97 plus 28, which of course is going to be 125. Actually, let me just write them all like that first. This one right over here is going to be 97 over 125. And of course, when you add this one and this one, it should add up to one. Likewise, this one's going to be 3... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
And of course, when you add this one and this one, it should add up to one. Likewise, this one's going to be 35 over 139, 35 plus 104, so 139. And this is going to be 104 over 104 plus 35, which is 139. And so let me just calculate each of them using this calculator. So let me scroll down a little bit. And so if I do 2... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
And so let me just calculate each of them using this calculator. So let me scroll down a little bit. And so if I do 28 divided by 125, I get 0.224. They said round your answers to the nearest hundredth. So this is 0.22. No accident within the last year, 97 divided by 125. So 97 divided by 125 is equal to, let's see, he... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
They said round your answers to the nearest hundredth. So this is 0.22. No accident within the last year, 97 divided by 125. So 97 divided by 125 is equal to, let's see, here if I round to the nearest hundredth, I'm going to round up 0.78. So this is 0.78. Then 35 divided by 139, 35 divided by 139 is equal to, round to... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
So 97 divided by 125 is equal to, let's see, here if I round to the nearest hundredth, I'm going to round up 0.78. So this is 0.78. Then 35 divided by 139, 35 divided by 139 is equal to, round to the nearest hundredth, 0.25. 0.25. And then 104 divided by 139. 104 divided by 139 gets me, if I round to the nearest hundre... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
0.25. And then 104 divided by 139. 104 divided by 139 gets me, if I round to the nearest hundredth, 0.75. 0.75. And I can check my answer, and I got it right. But the key thing here is to make sure we understand what's going on here. So one way to think about this is 22% of the sport utility vehicles had an accident wi... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
0.75. And I can check my answer, and I got it right. But the key thing here is to make sure we understand what's going on here. So one way to think about this is 22% of the sport utility vehicles had an accident within the last year, or you could say 0.22 of them. And you could say 78%, or 0.78, of the sport utility ve... | Two-way relative frequency tables Data and modeling 8th grade Khan Academy.mp3 |
So he took a random sample of 24 games and recorded their outcomes. Here are his results. So out of the 24 games, he won four, lost 13, and tied seven times. He wants to use these results to carry out a chi-squared goodness-of-fit test to determine if the distribution of his outcomes disagrees with an even distribution... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
He wants to use these results to carry out a chi-squared goodness-of-fit test to determine if the distribution of his outcomes disagrees with an even distribution. What are the values of the test statistic, the chi-squared test statistic, and p-value for Kenny's test? So pause this video and see if you can figure that ... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
Okay, so he's essentially just doing a hypothesis test using the chi-squared statistic because it's a hypothesis that's thinking about multiple categories. So what would his null hypothesis be? Well, his null hypothesis would be that all of the outcomes are equal probability. Outcomes equal, equal probability. And then... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
Outcomes equal, equal probability. And then his alternative hypothesis would be that his outcomes have not equal, not equal probability. Remember, we assume that the null hypothesis is true. And then assuming if the null hypothesis is true, the probability of getting a result at least this extreme is low enough, then w... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
And then assuming if the null hypothesis is true, the probability of getting a result at least this extreme is low enough, then we would reject our null hypothesis. Another way to think about it is if our p-value is below a threshold, we would reject our null hypothesis. And so what he did is he took a sample of 24 gam... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
Now, before we even calculate our chi-squared statistic and figure out what's the probability of getting a chi-squared statistic that large or greater, let's make sure we meet the conditions for inference for a chi-squared goodness-of-fit test. So you've seen some of them, but some of them are a little bit different. O... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
I'll write them up here. The random condition. And that would be that this is truly a random sample of games. And it tells us right here, he took a random sample of his 24 games, so we meet that condition. The second condition, when we're talking about chi-squared hypothesis testing, is the large counts, large counts c... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
And it tells us right here, he took a random sample of his 24 games, so we meet that condition. The second condition, when we're talking about chi-squared hypothesis testing, is the large counts, large counts condition. And this is an important one to appreciate. This is that the expected number of each category of out... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
This is that the expected number of each category of outcomes is at least equal to five. Now, you might say, hey, wait, wait, I only got four wins, or Kenny only got four wins out of his sample of 24. But that does not violate the large counts condition. Remember, what is the expected number of wins, losses, and ties? ... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
Remember, what is the expected number of wins, losses, and ties? Well, if you were assuming the null hypothesis, where the outcomes have equal probability, so the expected, the expected, I could write right over here, it would be that it's 1 3rd, 1 3rd, 1 3rd. And so 1 3rd of 24 is eight, eight and eight. That's what K... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
That's what Kenny would expect. And since, because all of these are at least equal to five, we meet the large counts condition. And then the last condition is the independence condition. If we aren't sampling with replacement, then we just have to feel good that our sample size is no more than 10% of the population. An... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
If we aren't sampling with replacement, then we just have to feel good that our sample size is no more than 10% of the population. And he can definitely play more than 240 games in his life, so we would assume that we meet that condition as well. And so with that out of the way, we can actually calculate our chi-square... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
And so, let's see, our chi-squared statistic is going to be equal to, so for each category, it's going to be the difference between the expected and what he got in that sample, squared divided by the expected. So the first category is wins. So that's going to be four minus eight, four minus eight squared over an expect... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
13 is how many he got, how many he lost, minus eight expected, squared over the number expected, plus he got seven ties, he would have expected eight squared, all of that over eight. And so let's see, what is this? Four minus eight is negative four. You square that, you get 16. 13 minus eight is five. You square that, ... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
You square that, you get 16. 13 minus eight is five. You square that, you get 25. Seven minus eight is negative one. Square that, you get one. And 16 divided by eight is going to be two. 25 divided by eight is going to be, let's see, that's 3 1 8, so that's 3.125. | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
Seven minus eight is negative one. Square that, you get one. And 16 divided by eight is going to be two. 25 divided by eight is going to be, let's see, that's 3 1 8, so that's 3.125. And then 1 8 is 0.125, 0.125. You add these together, so let's see, it's gonna be two plus 3.125, 5.125, plus another.125, so that's goin... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
25 divided by eight is going to be, let's see, that's 3 1 8, so that's 3.125. And then 1 8 is 0.125, 0.125. You add these together, so let's see, it's gonna be two plus 3.125, 5.125, plus another.125, so that's going to be 5.25. So our chi-squared statistic is 5.25. And now to figure out our p-value, our p-value is goi... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
So our chi-squared statistic is 5.25. And now to figure out our p-value, our p-value is going to be equal to the probability of getting a chi-squared statistic greater than or equal to 5.25. And you could use a chi-squared table for that. And we always have to think about our degrees of freedom. We have one, two, three... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
And we always have to think about our degrees of freedom. We have one, two, three categories. So our degrees of freedom is going to be one less than that, or three minus one, which is two. So our degrees of freedom is going to be equal to two. And that makes sense, because you know for a certain number of games, if you... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
So our degrees of freedom is going to be equal to two. And that makes sense, because you know for a certain number of games, if you know the number of wins, and you know the certain number of losses, you can figure out the number of ties. Or if you know any two of these categories, you can always figure out the third. ... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
So that's why you have two degrees of freedom. And so let's get out our chi-squared table. So we have two degrees of freedom. So we are in this row. And where is 5.25? So 5.25 is right over there. And so our probability is going to be between 0.10 and 0.05. | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
So we are in this row. And where is 5.25? So 5.25 is right over there. And so our probability is going to be between 0.10 and 0.05. So our p-value is going to be greater than 0.05 and less than 0.10. And so for example, if ahead of time, and he should have done this ahead of time, he set a significance level of 5%, and... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
And so our probability is going to be between 0.10 and 0.05. So our p-value is going to be greater than 0.05 and less than 0.10. And so for example, if ahead of time, and he should have done this ahead of time, he set a significance level of 5%, and our p-value here is greater than 5%, which we just saw, he would fail ... | Chi-square goodness-of-fit example AP Statistics Khan Academy.mp3 |
For the antibiotic to be sufficiently effective, it has to kill at least 90% of bacteria when applied to a harmful bacteria culture. She applied her antibiotic to a Petri dish full of harmful bacteria, waited for it to take effect, and then tried to estimate the percentage of dead bacteria in it. She took a random samp... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
Then she calculated the margin of error and found that the true percentage of dead bacteria is most likely to be above 90%. So what's happening over here, she's trying to figure out what percentage of the total population of bacteria died. And maybe there's something about this bacteria, maybe just when you look at it ... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
And so she decides to estimate the true percentage by sampling 300 individual bacterium, or I always forget the singular case, by sampling 300 bacteria. And then in her sample, she found that 94% of them were dead and then the margin of error tells us, because the margin of error says that it's unlikely, or that it's v... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
Now let's answer these questions. What type of statistical study did Alma use? Well, she used a, she's trying to estimate a parameter for a population, in this case, the parameter was the percentage of all of the bacteria that died, she couldn't observe that directly, so instead, she took a random sample of the bacteri... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
So this is, when you're using a random sample to generate a statistic which estimates a parameter for a population, that's a sample study. So she ran a sample study. Now the next question is, is the study appropriate for the statistical questions it's supposed to answer? So what was the question that she's trying to an... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
So what was the question that she's trying to answer? Well, at least the way it's written, it seems like she's trying to answer whether or not her antibiotic works, whether it's an effective antibiotic, whether it's capable of killing bacteria. And you might be tempted to say, okay, well look, it looks like it killed, ... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
Maybe it was caused by the plastic in the Petri dish. Maybe the air in the Petri dish was too cold or went bad, or maybe it was handled in a weird way, or maybe that bacteria was just a bad culture and it somehow just spontaneously died on its own. She can't say with confidence that it was definitely the antibiotic. In... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
In order for her to make that statement, she would have to run a proper experiment. She would have to have a control and a treatment group where everything is equal except for the treatment group has the treatment. So if she had two Petri dishes that were kept in the same conditions with the same lighting, the same air... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
So she would have had to run an experiment. The most appropriate statistical study, or the most appropriate study would have been a proper controlled experiment where you have a control group where they don't have the antibiotic and a treatment group where they do have the antibiotic. So let's see what are the choices ... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
So yes, because she's in a proper study, no, I don't like that answer. No, because she can't know for certain that the true percentage of dead bacteria is above 90%. Well, I'm not gonna click on that because even if she knew for certain that the true percentage of dead bacteria were 95%, she can't feel confident that i... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
Once again, it could be caused by the air conditioner, could have been caused by the Petri dish, could have been caused by the lighting in the room. So no, because the study didn't have a treatment and a control group. Yeah, I would go with that one right over there. Yes, because she found the antibiotic kills more tha... | Appropriate statistical study example Probability and Statistics Khan Academy.mp3 |
You're impatient. You want your frozen yogurt immediately. And so you decide to conduct a study. You want to figure out the probability of there being lines of different sizes. When you go to the frozen yogurt store after school, exactly at four o'clock p.m. So in your study, the next 50 times you observe, you go to th... | Constructing probability model from observations 7th grade Khan Academy.mp3 |
You want to figure out the probability of there being lines of different sizes. When you go to the frozen yogurt store after school, exactly at four o'clock p.m. So in your study, the next 50 times you observe, you go to the frozen yogurt store at four p.m., you make a series of observations. You observe the size of th... | Constructing probability model from observations 7th grade Khan Academy.mp3 |
You observe the size of the line. So let me make two columns here. Line size is the left column. And on the right column, let's say this is the number of times observed. So times, times observed. Observed. Alright, times observed. | Constructing probability model from observations 7th grade Khan Academy.mp3 |
And on the right column, let's say this is the number of times observed. So times, times observed. Observed. Alright, times observed. My handwriting is O-B-S-E-E-R-V-E-D. Alright, times observed. Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no ... | Constructing probability model from observations 7th grade Khan Academy.mp3 |
Alright, times observed. My handwriting is O-B-S-E-E-R-V-E-D. Alright, times observed. Alright, so let's first think about, okay, so you go and you say, hey look, I see no people in line, exactly, or you see no people in line, exactly 24 times. You see one person in line, exactly 18 times. And you see two people in lin... | Constructing probability model from observations 7th grade Khan Academy.mp3 |
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