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You could have done it other ways. You could have had a bar for have antibodies and another bar for not have antibodies, and then you could have segmented the bar chart by whether they are adults, children, or infants. But if you did that, that would have been trying to understand whether having antibodies or not havin... | Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3 |
But there is some information lost in this segmented bar chart. For example, we have lost the fact that we have sampled or we have tested a lot more adults than children and far more children than infants. So one way to incorporate that data back into a visualization to essentially show how many people you sampled in e... | Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3 |
So this is a mosaic plot right over here. And one way to think about it is we have just adjusted the width of each of these bars based on how many people we tested. So we tested 200 people. And so you can view this width right over here as being 200. And you can see that we tested 120 adults. So the width of this first... | Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3 |
And so you can view this width right over here as being 200. And you can see that we tested 120 adults. So the width of this first bar, I guess you could say, although now we're dealing with a mosaic plot, this width right over here would be 60% of this entire width, which you can see that it is. And then the children ... | Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3 |
And then the children are 60 of the 200 that we tested. And so this width right over here would be 60 over the entire 200, or it would be about 30% of the entire width. And we can see that we tested the fewest number of infants. And so this 20 right over here represents the 20 infants we tested. And the reason why this... | Mosaic plots and segmented bar charts Exploring two-variable data AP Statistics Khan Academy.mp3 |
In a previous video, we began to think about how we can use a regression line, and in particular, the slope of a regression line based on sample data, how we can use that in order to make inference about the slope of the true population regression line. In this video, we're going to think about what are the conditions ... | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
So to help us remember these conditions, you might want to think about the LINER acronym, L-I-N-E-R, and if it isn't obvious to you, this almost is linear. Liner, with an A, it would be linear, and this is valuable because remember, we're thinking about linear regression. So the L right over here actually does stand fo... | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
So actual linear relationship between X and Y. Now, in a lot of cases, you might just have to assume that this is going to be the case when you see it on an exam, like an AP exam, for example. They might say, hey, assume this condition is met. Oftentimes, they'll say, assume all of these conditions are met. They just w... | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
Oftentimes, they'll say, assume all of these conditions are met. They just want you to maybe know about these conditions, but this is something to think about. If the underlying relationship is nonlinear, well, then maybe some of your inferences might not be as robust. Now, the next one is one we have seen before when ... | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
Now, the next one is one we have seen before when we're talking about general conditions for inference, and this is the independence, independence condition, and there's a couple of ways to think about it. Either individual observations are independent of each other, so you could be sampling with replacement, or you co... | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
The normal condition, and once again, many times people will just say, assume it's been met, but let me actually draw a regression line, but do it with a little perspective, and I'm gonna add a third dimension. Let's say that's the x-axis, and let's say this is the y-axis, and the true population regression line looks ... | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
So that would be that normal distribution there, and then let's say for this x right over here, you would expect a normal distribution as well, so just like, just like this. So for given x, the distribution of y's should be normal. Once again, many times you'll just be told to assume that that has been met because it m... | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
Now the next condition is related to that, and this is the idea of having equal variance, equal variance, and that's just saying that each of these normal distributions should have the same spread for a given x. And so you could say equal variance, or you could even think about them having the equal standard deviation.... | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
Last but not least, and this is one we've seen many times, this is the random condition. And this is that the data comes from a well-designed random sample or some type of randomized experiment. And this condition we have seen in every type of condition for inference that we have looked at so far. So I'll leave you the... | Conditions for inference on slope More on regression AP Statistics Khan Academy.mp3 |
Management claimed that 10% of employees earned this rating but Jules suspected it was actually less common. She obtained an anonymous random sample of 10 ratings for employees on her team. She wants to use the sample data to test her null hypothesis that the true proportion is 10% versus her alternative hypothesis tha... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
Which conditions for performing this type of test did Jules' sample meet? And when they're saying which conditions, they are talking about the three conditions, the random condition, the normal condition, and we've seen these before, and the independence condition. So I will let you pause the video now and try to figur... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
All right, now let's work through this together. So let's just remind ourselves what we're going to do in a significance test. We have our null hypothesis, we have our alternative hypothesis. What we do is we look at the population, the population size, there's 40 employees on staff at this company. We take a sample, i... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
What we do is we look at the population, the population size, there's 40 employees on staff at this company. We take a sample, in Jules' case, she took a sample size of 10, and then we calculate a sample statistic, in this case it is the sample proportion, which is equal to, let's just call it p hat sub one. And then w... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
And just as a bit of review, a p value is the probability of getting a result at least as extreme as this one if we assume our null hypothesis is true. And in this particular case, because she suspects that not 10% are getting the exceeds expectations, this would be the probability of your sample statistic being less t... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
But then if the p value is not less than this, then you wouldn't be able to reject the null hypothesis. But the key thing, and this is what this question is all about, in order to feel good about this calculation, we need to make some assumptions about the sampling distribution. We have to assume that it's reasonably n... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
The first is the random condition, and that's that the data points in this sample were truly randomly selected. So pause this video. Did she meet the random condition? Well, it says she obtained an anonymous random sample of 10 ratings of employees on her team. They don't say how she did it, but we'll have to take thei... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
Well, it says she obtained an anonymous random sample of 10 ratings of employees on her team. They don't say how she did it, but we'll have to take their word for it that it was an anonymous random sample, so she meets the random condition. Now, what about the normal condition? The normal condition tells us that the ex... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
The normal condition tells us that the expected number of successes, which would be our sample size times the true proportion, and the number of failures, sample size times one minus p, need to be at least equal to 10. So they need to be greater than or equal to 10. Now, what are they for this particular scenario? Well... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
Well, n is equal to 10, n is equal to 10, and our true proportion, remember, we're going to assume, when we do the significance test, we assume the null hypothesis is true, and the null hypothesis tells us that our true proportion is 0.1. So this is 0.1. This is one minus 0.1, which is 0.9. Well, 10 times 0.1 is one, s... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
Well, 10 times 0.1 is one, so that's not greater than or equal to 10, so just off of that, we don't meet the normal condition, but even the second one, 10 times 0.9 is nine. That's also not greater than or equal to 10, so we don't meet this normal condition. We can't feel good that the sampling distribution is roughly ... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
And then last but not least, independence. Independence is to feel good that each of the data points in your sample are independent. The results of whether they are a success or a failure is independent of each other. Now, if she was surveying these people with replacement, if each data point was with replacement, you ... | Conditions for a z test about a proportion AP Statistics Khan Academy.mp3 |
He suspects that on average, they send each other more than 100 messages per day. Sunil takes a random sample of seven days from their chat history and records how many messages were sent on those days. The sample data are strongly skewed to the right with a mean of 125 messages and a standard deviation of 44 messages.... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
He wants to use these sample data to conduct a t-test about the mean. Which conditions for performing this type of significance test have been met? So let's just think about what's going on here. Sunil might have some type of a null hypothesis. Maybe he got this 100, maybe he read a magazine article that says that on a... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
Sunil might have some type of a null hypothesis. Maybe he got this 100, maybe he read a magazine article that says that on average, the average teenager sends 100 text messages per day. And so maybe the null hypothesis is that the mean amount of messages per day that he and his friends send, which was signified by mu, ... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
And so what he does is he takes a sample from the population of days, and there's over 365. They say they've been using the group messaging app for over a year. And he takes seven of those days. So n is equal to seven. And from that, he calculates sample statistics. He calculates the sample mean, which is trying to est... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
So n is equal to seven. And from that, he calculates sample statistics. He calculates the sample mean, which is trying to estimate the true population mean right over here. And he also is able to calculate a sample standard deviation. And what you do in a significance test is you say, well, what is the probability of g... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
And he also is able to calculate a sample standard deviation. And what you do in a significance test is you say, well, what is the probability of getting this sample mean or something even more extreme, assuming the null hypothesis? And if that probability is below a preset threshold, then you would reject the null hyp... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
But in order to feel good about that significance test and be able to even calculate that p-value with confidence, there are conditions for performing this type of significance test. The first is is that this is truly a random sample, and that's known as the random condition. And you have seen this before when we did s... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
Here we're doing it with means, population mean, sample mean. In the past, we did it with population proportion and sample proportion. Well, the random condition, it says it right here, Sunil takes a random sample of seven days from their chat history. They don't say how he did it, but we'll just take their word for it... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
They don't say how he did it, but we'll just take their word for it that it was a random sample. The next condition is sometimes known as the independence, independence condition. And that's that the individual observations in our sample are roughly independent. One way that they would be independent for sure is if Sun... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
One way that they would be independent for sure is if Sunil is sampling with replacement. They don't say that, but another condition, so you either could have replacement, sampling with replacement, or another way where you could feel that it's roughly independent is if your sample size is less than or equal to 10% of ... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
So seven is for sure less than or equal to 10% of 365, which would be 36.5. So we meet this condition, which allows us to meet the independence condition. Now the last condition is often known as the normal condition, and this is to feel good that the sampling distribution of the sample means right over here is approxi... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
And this is going to be a little bit different than what we saw with significance tests when we dealt with proportions. There's a few ways to feel good that the sampling distribution of the sample means is normal. One is is if the underlying parent population normal. So parent, parent population normal. Now they don't ... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
So parent, parent population normal. Now they don't tell us anything that there's actually a normal distribution for the amount of time that they spend on a given day. So we don't know this one for sure, but sometimes you might. Another way is to feel good that our sample size is greater than or equal to 30. And this c... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
Another way is to feel good that our sample size is greater than or equal to 30. And this comes from the central limit theorem that then our sampling distribution is going to be roughly normal. But we see very clearly our sample size is not greater than or equal to 30, so we don't meet that constraint either. Now the t... | Conditions for a t test about a mean AP Statistics Khan Academy.mp3 |
Darnell is a middle school student with a height of 161.4 centimeters. What proportion of student heights are lower than Darnell's height? So let's think about what they are asking. So they're saying that heights are normally distributed. So it would have a shape that looks something like that. That's my hand-drawn ver... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So they're saying that heights are normally distributed. So it would have a shape that looks something like that. That's my hand-drawn version of it. There's a mean of 150 centimeters. So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a ... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
There's a mean of 150 centimeters. So right over here, that would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters, and Darnell has a height of 161.4 centimeters. So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. T... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So Darnell is above the mean. So let's say he is right over here, and I'm not drawing it exactly, but you get the idea. That is 161.4 centimeters. And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over h... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
And we want to figure out what proportion of students' heights are lower than Darnell's height. So we want to figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a hint on how to do this. We need to think about how ... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
And I'll give you a hint on how to do this. We need to think about how many standard deviations above the mean is Darnell. And we can do that because they tell us what the standard deviation is, and we know the difference between Darnell's height and the mean height. And then once we know how many standard deviations h... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
And then once we know how many standard deviations he is above the mean, that's our z-score, we can look at a z-table that tell us what proportion is less than that amount in a normal distribution. So let's do that. So I have my TI-84 emulator right over here. And let's see, Darnell is 161.4 centimeters, 161.4. Now the... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
And let's see, Darnell is 161.4 centimeters, 161.4. Now the mean is 150, minus 150, is equal to, we could have done that in our head, 11.4 centimeters. Now how many standard deviations is that above the mean? Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
Well they tell us that a standard deviation in this case for this distribution is 20 centimeters. So we'll take 11.4 divided by 20. So we will just take our previous answer. So this just means our previous answer divided by 20 centimeters. And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 ... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So this just means our previous answer divided by 20 centimeters. And that gets us 0.57. So we can say that this is 0.57 standard deviations 0.57 standard deviations, deviations above the mean. Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion i... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
Now why is that useful? Well you could take this z-score right over here and look at a z-table to figure out what proportion is less than 0.57 standard deviations above the mean. So let's get a z-table over here. So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, th... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So what we're going to do is we're gonna look up this z-score on this table. And the way that you do it, this first column, each row tells us our z-score up until the tenths place. And then each of these columns after that tell us which hundredths we're in. So 0.57, the tenths place is right over here. So we're going t... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
So 0.57, the tenths place is right over here. So we're going to be in this row. And then our hundredths place is this seven. So we'll look right over here. So 0.57, this tells us the proportion that is lower than 0.57 standard deviations above the mean. And so it is 0.7157, or another way to think about it is, if the h... | Standard normal table for proportion below AP Statistics Khan Academy.mp3 |
It is election season and there is a runoff between candidate A versus candidate B. And we are pollsters and we're interested in figuring out well what's the likelihood that candidate A wins this election? Well ideally, we would go to the entire population of likely voters right over here. Let's say there's 100,000 lik... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Let's say there's 100,000 likely voters and we would ask every one of them, who do you support? And from that, we would be able to get the population proportion. Which would be, this is the proportion that support candidate A. But it might not be realistic. In fact, it definitely will not be realistic to ask all 100,00... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
But it might not be realistic. In fact, it definitely will not be realistic to ask all 100,000 people. So instead, we do the thing that we tend to do in statistics is that we sample this population and we calculate a statistic from that sample in order to estimate this parameter. So let's say we take a sample right ove... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So let's say we take a sample right over here. So this sample size, let's say n equals 100 and we calculate the sample proportion that support candidate A. So out of the 100, let's say that 54 say that they're going to support candidate A. So the sample proportion here is 0.54. And just to appreciate that we're not alw... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So the sample proportion here is 0.54. And just to appreciate that we're not always going to get 0.54, there could have been a situation where we sampled a different 100 and we would have maybe gotten a different sample proportion, maybe in that one we got 0.58. And we already have the tools in statistics to think abou... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
We've talked about it when we thought about sampling distributions. So you could have the sampling distribution of the sample proportions, of the sample proportions, and it's going to, this distribution is going to be specific to what our sample size is for n is equal to 100. And so we can describe the possible sample ... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So let me do that. So it would look something like this. Because our sample size is so much smaller than the population, it's way less than 10%, we can assume that each person we're asking that it's approximately independent. Also, if we make the assumption that the true proportion isn't too close to zero or not too cl... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Also, if we make the assumption that the true proportion isn't too close to zero or not too close to one, then we can say that, well look, this sampling distribution is roughly going to be normal. So we'll have a normal, this kind of bell curve shape. And we know a lot about the sampling distribution of the sample prop... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
We know already, for example, and if this is foreign to you, I encourage you to watch the videos on this on Khan Academy, that the mean of this sampling distribution is going to be the actual population proportion. And we also know what the standard deviation of this is going to be. So let me, that's maybe, that's one ... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Above the mean, that's one standard deviation, two standard deviations, three standard deviations below the mean. So this distance, let me do this in a different color. This standard deviation right over here, which we denote as the standard deviation of the sample proportions for this sampling distribution. This is, w... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
This is, we've already seen the formula there. It's the square root of p times one minus p, where p is once again our population proportion, divided by our sample size. That's why it's specific for n equals 100 here. And so in this first scenario, and let's just focus on this one right over here. When we took a sample ... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And so in this first scenario, and let's just focus on this one right over here. When we took a sample size of n equals 100 and we got the sample proportion of 0.54, we could have gotten all sorts of outcomes here. Maybe 0.54 is right over here. Maybe 0.54 is right over here. And the reason why I have this uncertainty ... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Maybe 0.54 is right over here. And the reason why I have this uncertainty is we actually don't know what the real population parameter is, what the real population proportion is. But let me ask you maybe a slightly easier question. What is, what is the probability, probability that our sample proportion of 0.54 is with... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
What is, what is the probability, probability that our sample proportion of 0.54 is within, is within two times two standard deviations of p? Pause the video and think about that. Well, that's just saying, look, if I'm gonna take a sample and calculate the sample proportion right over here, what's the probability that ... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Well, that's essentially going to be this area right over here. And we know from studying normal curves that approximately 95% of the area is within two standard deviations. So this is approximately 95%. 95% of the time that I take a sample size of 100 and I calculate this sample proportion, 95% of the time I'm going t... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
95% of the time that I take a sample size of 100 and I calculate this sample proportion, 95% of the time I'm going to be within two standard deviations. But if you take this statement, you can actually construct another statement that starts to feel a little bit more, I guess we could say, inferential. We could say the... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Pause this video. Appreciate that these two are equivalent statements. If there's a 95% chance that our sample proportion is within two standard deviations of the true proportion, well, that's equivalent to saying that there's a 95% chance that our true proportion is within two standard deviations of our sample proport... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And this is really, really interesting because if we were able to figure out what this value is, well, then we would be able to create what you could call a confidence interval. Now, you immediately might be seeing a problem here. In order to calculate this, our standard deviation of this distribution, we have to know ... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So pause this video and think about what we would do instead if we don't know what p is here, if we don't know our population proportion, do we have something that we could use as an estimate for our population proportion? Well, yes, we calculated p hat already. We calculated our sample proportion. And so a new statist... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And so a new statistic that we could define is the standard error. The standard error of our sample proportions. And we can define that as being equal to, since we don't know the population proportion, we're gonna use a sample proportion. P hat times one minus p hat, all of that over n. In this case, of course, n is 10... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
P hat times one minus p hat, all of that over n. In this case, of course, n is 100. We do know that. And it actually turns out, I'm not gonna prove it in this video, that this actually is an unbiased estimator for this right over here. So this is going to be equal to 0.54 times one minus 0.54. So it's 0.46. All of that... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So this is going to be equal to 0.54 times one minus 0.54. So it's 0.46. All of that over 100. So we have the square root of 0.54 times 0.46 divided by 100, close my parentheses, Enter. So if we're around to the nearest hundredth, it's going to be actually, even around to the nearest thousandth, it's gonna be approxima... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So we have the square root of 0.54 times 0.46 divided by 100, close my parentheses, Enter. So if we're around to the nearest hundredth, it's going to be actually, even around to the nearest thousandth, it's gonna be approximately five hundredths. So this is going to be, this is approximately 0.05. So another way to say... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So another way to say all of these things is, instead, we don't know exactly this, but now we have an estimate for it. So we can now say, with 95% confidence, and that will often be known as our confidence level right over here, with 95% confidence, between, between, and so we'd wanna go two standard errors below our s... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So that would be 0.54 minus 10 hundredths, which would be 0.44. And we'd also wanna go two standard errors above the sample proportion. So that would be that plus 10 hundredths, and 0.64 of voters, of voters, support, support A. And so this interval that we have right over here, from 0.44 to 0.64, this will be known as... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
And so this interval that we have right over here, from 0.44 to 0.64, this will be known as our confidence interval. Confidence interval. And this will change, not just in the starting point and the end point, but it will change the actual length of our confidence interval, will change depending on what sample proporti... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
A related idea to the confidence interval is this notion of margin of error. Margin of error. And for this particular case, for this particular sample, our margin of error, because we care about 95% confidence, so that would be two standard errors. So our margin of error here is two times our standard error, which is b... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
So our margin of error here is two times our standard error, which is be 0.1 or 0.10. And so we're going one margin of error above our sample proportion, right over here, and one margin of error below our sample proportion, right over here, to define our confidence interval. And as I mentioned, this margin of error is ... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
Depending on what our sample proportion is, it's going to affect our margin of error, because that is calculated essentially with the standard error. Another interpretation of this is that the method that we used to get this interval right over here, the method that we used to get this confidence interval, when we use ... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
I'll cover that intuition more in future videos. We'll see how the interval changes, how the margin of error changes, but when you do this calculation over and over and over again, 95% of the time, your true proportion is going to be contained in whatever interval you happen to calculate that time. Now another interest... | Confidence intervals and margin of error AP Statistics Khan Academy.mp3 |
His sample mean was four years and his sample standard deviation was two years. Rory wants to use these sample data to conduct a t-test on the mean. Assume that all conditions for inference have been met. Calculate the test statistic for Rory's test. So I always just like to remind ourselves what's going on. So you hav... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
Calculate the test statistic for Rory's test. So I always just like to remind ourselves what's going on. So you have your null hypothesis here that the mean number of years of experience for teachers in the district is five and then the alternative hypothesis is that the mean years of experience is less than five for t... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
So if this represents all the teachers in the district, the population, then what he did is he took a sample and said he used a sample of 25 teachers. So n here is equal to 25. And then from that sample, he was able to calculate some statistics. He was able to calculate the sample mean. So that sample mean was four yea... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
He was able to calculate the sample mean. So that sample mean was four years. The sample mean was four years. And then he was also able to calculate the sample standard deviation. The sample standard deviation was equal to two years. Now, the whole point that we do or the main thing we do when we do significance tests ... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
And then he was also able to calculate the sample standard deviation. The sample standard deviation was equal to two years. Now, the whole point that we do or the main thing we do when we do significance tests is we say, all right, if we assume the null hypothesis is true, what's the probability of getting a sample mea... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
And if that probability is below a preset significance level, then we reject the null hypothesis and it suggests the alternative. But in order to figure out that probability, we need to figure out a test statistic. Sometimes we use a z-test. If we're dealing with proportions. But when we deal with means, we tend to use... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
If we're dealing with proportions. But when we deal with means, we tend to use a t-test. And the reason why is if you wanted to figure out a z-statistic, what you would do is you would take your sample mean, subtract from that the assumed mean from the null hypothesis. So mu, and I'll just put a little zero, sub zero t... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
So mu, and I'll just put a little zero, sub zero there. So this is the assumed mean from the null hypothesis. And then you would want to divide by the standard deviation of the sampling distribution of the sample mean. So you'd wanna divide by that. But this, we don't know. And so that's why instead we do a t-statistic... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
So you'd wanna divide by that. But this, we don't know. And so that's why instead we do a t-statistic, in which case we take the difference between our sample mean and our assumed population mean, the population parameter, and we try to estimate this. And we estimate that with our sample standard deviation divided by t... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
And we estimate that with our sample standard deviation divided by the square root of our sample size. And so if you're inspired, I encourage you, even if you're not inspired, I encourage you to pause this video and try to calculate this t-statistic. Well, this is going to be equal to, let's see, our sample mean is fou... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
Our sample standard deviation is two. All of that over the square root of the sample size, all of that over the square root of 25. So this is going to be equal, our numerator is negative one. So it's negative one divided by two over five, which is equal to negative one times five over two. And so this is going to be eq... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
So it's negative one divided by two over five, which is equal to negative one times five over two. And so this is going to be equal to, equal to negative five over two, or negative 2.5. And then what we would do in this, what Rory would do, is then look this t-value up on a t-table and say, so if you look at a distribu... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
So negative, negative 2.5. And so what he would wanna do is figure out this area here, because this would be the probability of being that far below the mean or even further below the mean. And so that would give us our p-value. And then if that p-value is below some preset significance level that Rory should have set,... | Example calculating t statistic for a test about a mean AP Statistics Khan Academy.mp3 |
What is the probability that you will run out of water? So let's think about what's happening here. So there's some distribution of how many liters the average man needs when they're active outdoors. And let me just draw an example. It might look something like this. So they're all going to need at least more than 0 li... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
And let me just draw an example. It might look something like this. So they're all going to need at least more than 0 liters. So this would be 0 liters over here. The average male, so the mean of the amount of water a man needs when active outdoors, is 2 liters. So 2 liters would be right over here. So the mean is equa... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So this would be 0 liters over here. The average male, so the mean of the amount of water a man needs when active outdoors, is 2 liters. So 2 liters would be right over here. So the mean is equal to 2 liters. It has a standard deviation of 0.7 liters, or 0.7 liters. So the standard deviation, maybe I'll draw it this wa... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
So the mean is equal to 2 liters. It has a standard deviation of 0.7 liters, or 0.7 liters. So the standard deviation, maybe I'll draw it this way. So this distribution, once again, we don't know whether it's a normal distribution or not. It could just be some type of crazy distribution. So maybe some people need almos... | Sampling distribution example problem Probability and Statistics Khan Academy.mp3 |
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