Sentence stringlengths 102 3.15k | video_title stringlengths 51 104 |
|---|---|
And it is not skewed to the right or skewed to the left like this. And so right here it says, look, the sample size times our sample proportion has to be greater than or equal to 10. Or our sample size times one minus our sample proportion has to be greater than or equal to 10. Well, another way to think about this is ... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
Well, another way to think about this is our successes, our successes in our sample need to be greater than or equal to 10. And our failures need to be greater than or equal to 10. Well, how many successes were there? There were seven, seven. And you could even say, look, our n is 30 times our sample proportion is seve... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
There were seven, seven. And you could even say, look, our n is 30 times our sample proportion is seven over 30, which is going to be seven. So our successes is less than 10. So actually we violate the normal condition. And once again, this is a rule of thumb, but this is telling us that our actual sampling distributio... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
So actually we violate the normal condition. And once again, this is a rule of thumb, but this is telling us that our actual sampling distribution might be skewed. Remember, this is just based on one sample, what we're able to figure out. This is one sample z interval. We might be wrong, but we wouldn't feel good that ... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
This is one sample z interval. We might be wrong, but we wouldn't feel good that we're meeting the normal condition here. So I would rule this one out. Individual observations can be considered independent. Well, if he randomly selected people with replacement, then they could be independent. Or if the people he is sel... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
Individual observations can be considered independent. Well, if he randomly selected people with replacement, then they could be independent. Or if the people he is selecting, if his sample size is less than 10% of the total population, then it could be considered independent, even though it wouldn't be perfectly indep... | Conditions for confidence intervals worked examples AP Statistics Khan Academy.mp3 |
Adriana gathered data on different schools' winning percentages and the average yearly salary of their head coaches in millions of dollars in the years 2000 to 2011. She then created the following scatter plot and trend line. So this is salary in millions of dollars and the winning percentage. And so here we have a coa... | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
And so here we have a coach who made over $4 million and it looks like they won over 80% of their games. Then you have this coach over here who has a salary of a little over a million and a half dollars and they are winning over 85%. And so each of one of these data points is a coach and it's plotting their salary or t... | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
Assuming the line correctly shows the trend in the data and it's a bit of an assumption. There are some outliers here that are well away from the model and this isn't a, it looks like there's a linear, a positive linear correlation here, but it's not super tight and there's a bunch of coaches right over here in the low... | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
Well if you believe the model, then a y-intercept of being 39 would be, the model is saying that if someone makes no money, that they could, zero dollars, that they could win, that the model would expect them to win 39% of their gains, which seems a little unrealistic because you would expect most coaches to get paid s... | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
The average salary was $39 million. No, no one on our chart made 39 million. On average, each million dollar increase in salary was associated with a 39% increase in winning percentage. So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%. No, that... | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
So that would be something related to the slope, and the slope was definitely not 39. The average winning percentage was 39%. No, that wasn't the case either. The model indicates that teams with coaches who had a salary of zero million dollars will average a winning percentage of approximately 39%. Yeah, this is the cl... | Interpreting y-intercept in regression model AP Statistics Khan Academy.mp3 |
So it says here I have a 0.35 probability of making a free throw. What is the probability of making four out of seven free throws? Well, this is a classic binomial random variable question. If we said the binomial random variable X is equal to number of made free throws from seven, I could say seven trials or seven sho... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
If we said the binomial random variable X is equal to number of made free throws from seven, I could say seven trials or seven shots, seven trials with the probability of success is equal to 0.35 for each free throw. So really this question amounts to what is the probability that my binomial random variable X is equal ... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
And what you're going to want to do here is use three arguments. So the first one is the number of trials. So in this case it is seven. And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make ... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
And if you're doing it on the free response section of the AP test, you should make it very clear that that right over there is your N. The graders will actually look for that to make sure that you're not just guessing what goes where. So you would say that is my N and then you would say your probability, 0.35. And onc... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
That is your P. And then last but not least, what is the probability that a binomial variable, when you're taking seven trials with a probability of success of each of them being 0.35, that you have exactly four successes? So now let's get our calculator out and actually do that. All right, so now we have our graphing ... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
So there's a couple of ways to input this. You could just type it in directly. That could take time. You could do second and this little blue distribution here. So there you have it. In order to get to the function, you could either scroll down or you could scroll up to get to the bottom of the list and you see it righ... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
You could do second and this little blue distribution here. So there you have it. In order to get to the function, you could either scroll down or you could scroll up to get to the bottom of the list and you see it right over here, binomPDF. You could do alpha A to go there really fast or you could just scroll up here,... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
You could do alpha A to go there really fast or you could just scroll up here, click Enter. And then you have the number of trials that you wanna deal with. Well, we're gonna take seven trials. The probability of success in each trial is 0.35. And then my X value, well, I wanna find the probability that my binomial ran... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
The probability of success in each trial is 0.35. And then my X value, well, I wanna find the probability that my binomial random variable is equal to four, four successes out of the trials. And now let me go to Paste and this is actually going to type in exactly what we had before. Notice this is the exact same thing.... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
Notice this is the exact same thing. So I have seven trials, P is equal to 0.35 and I wanna know the probability of having exactly four successes. And then I just click Enter and I get, there you go, 0.14. So this is equal to approximately 0.14. Now based on the same binomial random variable, if we're then asked what i... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
So this is equal to approximately 0.14. Now based on the same binomial random variable, if we're then asked what is the probability of making less than five free throws? So we could say this is the probability that X is less than five or we could say this is the probability that X is less than or equal to four. And the... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
And the reason why I write it this way is because using it this way, you can now use the binomial cumulative distribution function on my calculator. So if I just type in binom, and once again, I'm gonna take seven, or binom CDF, I should say, cumulative distribution function, and I'm gonna take seven trials and the pro... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
It is the probability that I make zero, one, two, three, or four free throws. So all of the possible outcomes of my binomial random variable up to and including this value right over here. So let me get that, let me get my calculator back. So once again, I can go to second, distribution. I'll scroll up to go to the bot... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
So once again, I can go to second, distribution. I'll scroll up to go to the bottom of the list and here you see it, binomial cumulative distribution function. So let me go there, click enter. And once again, seven trials. My P is 0.35. And my X value is four, but now this is gonna give me the probability that my binom... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
And once again, seven trials. My P is 0.35. And my X value is four, but now this is gonna give me the probability that my binomial random variable equals four. This is going to give me the probability that I get any value up to and including four. So this should be a higher probability. And there you have it. It is app... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
This is going to give me the probability that I get any value up to and including four. So this should be a higher probability. And there you have it. It is approximately 0.94. So this is approximately 0.94. So hopefully you found that helpful. These calculators can be very useful, especially on something like an AP St... | Binompdf and binomcdf functions Random variables AP Statistics Khan Academy.mp3 |
And homogeneity or homogeneity, in everyday language, this means how similar things are. And that's what we're essentially going to test here. We're gonna look at two different groups and see whether the distributions of those groups for a certain variable are similar or not. And so the question I'm going to think abou... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And so the question I'm going to think about or we're going to think about together in this video is, let's say we were thinking about left-handed versus right-handed people. And we're wondering, do they have the same preferences for subject domains? Are they equally inclined to science, technology, engineering, math, ... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And so we can set up our null and alternative hypotheses. Our null hypothesis is that there is no difference in the distribution between left-handed and right-handed people in terms of their preference for subject domains. So no difference in subject, subject preference for left and right, for left and right-handed fol... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And then the alternative hypothesis, well, no, there is a difference. So there is a difference. So how would we go about testing this? Well, we've done hypothesis testing many times in many videos already. But here, we're going to sample from two different groups. So let's say that this is the population of right-hande... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
Well, we've done hypothesis testing many times in many videos already. But here, we're going to sample from two different groups. So let's say that this is the population of right-handed folks, and this is the population of left-handed folks. Let's say from that sample of right-handed folks, I take a sample of 60, and ... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
Let's say from that sample of right-handed folks, I take a sample of 60, and then I do the same thing for the left-handed folks. And these don't even have to be the same sample sizes. So the left-handed folks, let's say I sample 40 folks. And here is the data that I actually collect. So for those 60 right-handed folks,... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And here is the data that I actually collect. So for those 60 right-handed folks, 30 of them preferred the STEM subjects, science, technology, engineering, math. 15 preferred humanities. And 15 were indifferent. They liked them equally. And then for the 40 left-handed folks, I got 10 preferring STEM, 25 preferring huma... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And 15 were indifferent. They liked them equally. And then for the 40 left-handed folks, I got 10 preferring STEM, 25 preferring humanities, and five viewed them equally. And then you see the total number of right-handed folks, total number of left-handed folks. And then you have the total number from both groups that ... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And then you see the total number of right-handed folks, total number of left-handed folks. And then you have the total number from both groups that preferred STEM, total number from both groups that preferred humanities, total number from both groups that had no preference. So let's just start thinking about what the ... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
This is the right-handed column. This is the left-handed column. Well, assuming that the null hypothesis is true, that there's no difference between right and left-handed people in terms of their preference, our best estimate of what the distribution of preference would be in the population generally would come from th... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
Since we're assuming no difference, we would assume that in either group, 40 out of every 100 would prefer STEM, or 40%. 40% would prefer humanities. And 20% would have no preference. And so our expected would be that 40% of the 60 right-handed folks would prefer STEM. So what's 40% of 60? .4 times 60 is 24. And simila... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And so our expected would be that 40% of the 60 right-handed folks would prefer STEM. So what's 40% of 60? .4 times 60 is 24. And similarly, we would expect 40% preferring humanities. 40% times 60 is 24 again. And then we would expect 20% of the right-handed group to have no preference. So 20% of 60 is 12. | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And similarly, we would expect 40% preferring humanities. 40% times 60 is 24 again. And then we would expect 20% of the right-handed group to have no preference. So 20% of 60 is 12. And these, once again, they add up to 60. And then for the left-handed folks, we would go through the same process. We would expect that 4... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
So 20% of 60 is 12. And these, once again, they add up to 60. And then for the left-handed folks, we would go through the same process. We would expect that 40% of them prefer STEM. 40% of 40, that is 16. On the humanities, again, 40% of 40 is 16. And equal, 20% of 40 is eight. | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
We would expect that 40% of them prefer STEM. 40% of 40, that is 16. On the humanities, again, 40% of 40 is 16. And equal, 20% of 40 is eight. And then all of these add up to 40. Once you calculate these expected values, it's a good time to make sure you're meeting your conditions for conducting a chi-squared test. The... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And equal, 20% of 40 is eight. And then all of these add up to 40. Once you calculate these expected values, it's a good time to make sure you're meeting your conditions for conducting a chi-squared test. The first is the random condition. And so these need to be truly random samples. So hopefully we met that condition... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
The first is the random condition. And so these need to be truly random samples. So hopefully we met that condition. The second is that the expected value for any of these data points have to be at least equal to five. And so we have met that condition. These are all at least equal to five. And then the last condition ... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
The second is that the expected value for any of these data points have to be at least equal to five. And so we have met that condition. These are all at least equal to five. And then the last condition is the independence condition that we are either sampling with replacement, or if we're not sampling with replacement... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And then the last condition is the independence condition that we are either sampling with replacement, or if we're not sampling with replacement, we have to feel good that our samples are no more than 10% of the population. So let's assume that that is the case as well. And now we're ready to calculate our chi-squared... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
We would get our chi-squared statistic is going to be equal to the difference between what we got and the expected squared, so 30 minus 24 squared, divided by the expected, divided by 24. And we'll do it for all six of these data points. So then I will go to the next one. So then this is going to be, so plus, and if I ... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
So then this is going to be, so plus, and if I look at this and this here, I'm going to have 10 minus 16 squared over expected, 16. And then I'm going to have, I'll look at that data point and that expected, and I would get 15 minus 24 squared over expected, over 24. I'm running out of colors. And then we would look at... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And then we would look at that, those two numbers, and we would say plus 25 minus 16 squared divided by expected. And then we would get, we would look at these two, plus 15 minus 12 squared over expected, over 12. And then last but not least, let me find a color I haven't used, we would look at that and that, and we wo... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
Now once you get that value for the chi-square statistic, the next question is what are the degrees of freedom? Now a simple rule of thumb is to just look at your data and think about the number of rows and the number of columns. And we have three rows and two columns. And so your degrees of freedom are going to be the... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And so your degrees of freedom are going to be the number of rows minus one, three minus one, times the number of columns minus one, two minus one. And so this is going to be equal to two times one, which is equal to two. Now the reason why that makes intuitive sense is, think about it, if you knew two of these data po... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
If you knew these two data points, you could figure out that. If you knew this data point and you knew the total, you could figure out that. If you knew this data point and you knew the total, you could figure out that. And if you figured out that and that, then you could figure out this right over here. And so that's ... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
And if you figured out that and that, then you could figure out this right over here. And so that's why this rule of thumb works. The number of rows minus one times the number of columns minus one gives you your degrees of freedom. Now, given this chi-squared statistic that I haven't calculated, but you could type this... | Introduction to the chi-square test for homogeneity AP Statistics Khan Academy.mp3 |
Alright, this is the telltale signs of a geometric random variable. How many trials do I have to take until I get a success? Let m be the number of shots it takes Jeremiah to successfully make his first three-point shot. Okay, so they're defining the random variable here, the number of shots it takes, the number of tri... | Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3 |
Okay, so they're defining the random variable here, the number of shots it takes, the number of trials it takes until we get a successful three-point shot. Assume that the results of each shot are independent. Alright, the probability that he makes a given shot is not dependent on whether he made or missed the previous... | Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3 |
Find the probability that Jeremiah's first successful shot occurs on his third attempt. So like always, pause this video and see if you can have a go at it. Alright, now let's work through this together. So we wanna find the probability that, so m is the number of shots it takes until Jeremiah makes his first successfu... | Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3 |
So we wanna find the probability that, so m is the number of shots it takes until Jeremiah makes his first successful one. And so what they're really asking is find the probability that m is equal to three, that his first successful shot occurs on his third attempt. So m is equal to three. So that the number of shots i... | Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3 |
So that the number of shots it takes Jeremiah to make his first successful shot is three. So how do we do this? Well, what's just the probability of that happening? Well, that means he has to miss his first two shots and then make his third shot. So what's the probability of him missing his first shot? Well, if he has ... | Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3 |
Well, that means he has to miss his first two shots and then make his third shot. So what's the probability of him missing his first shot? Well, if he has a 1 4th chance of making his shots, he has a 3 4th probability of missing his shots. So this will be 3 4ths, so he misses the first shot, times he has to miss the se... | Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3 |
So this will be 3 4ths, so he misses the first shot, times he has to miss the second shot, and then he has to make his third shot. So there you have it, that's the probability. Miss, miss, make. And so what is this going to be? This is equal to nine over 60 4ths. So there you have it. If you wanted to have this as a de... | Probability for a geometric random variable Random variables AP Statistics Khan Academy.mp3 |
20 teachers were asked about their favorite course. Seven teachers said language, three teachers said history, nine teachers said geometry, one teacher said chemistry, zero teachers said physics. Create a bar chart showing everyone's favorite courses. So we've got the bar chart right over here, and let's see what we ne... | Creating a bar chart Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So we've got the bar chart right over here, and let's see what we need to plot. So it said zero teachers said physics, which is surprising to me, because since physics is arguably my favorite course. But let's plot what the data has. So physics, so right now it looks like it's halfway between zero and one, so I actuall... | Creating a bar chart Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
So physics, so right now it looks like it's halfway between zero and one, so I actually have to bring the physics down to zero. See chemistry, they said, let's see, one teacher said chemistry, so we gotta bring chemistry up to one. Now, nine teachers said geometry. So geometry, let's bring that up to nine. One teacher ... | Creating a bar chart Applying mathematical reasoning Pre-Algebra Khan Academy.mp3 |
One of the most commonly used tools in all of statistics is the notion of a z-score. And one way to think about a z-score is it's just the number of standard deviations away from the mean that a certain data point is. So let me write that down. Number of standard deviations, I'll write it like this. Number of standard ... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
Number of standard deviations, I'll write it like this. Number of standard deviations from our population mean for a particular, particular data point. Now let's make that a little bit concrete. Let's say that you're some type of marine biologist and you've discovered a new species of winged turtles. And there's a tota... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
Let's say that you're some type of marine biologist and you've discovered a new species of winged turtles. And there's a total of seven winged turtles. The entire population of these winged turtles is seven. And so you go and you're actually able to measure all the winged turtles. So, and you care about their length. A... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
And so you go and you're actually able to measure all the winged turtles. So, and you care about their length. And you also wanna care about how are those lengths distributed? Lengths of winged turtles. All right. And let's say, and this is all in centimeters. These are very small turtles. | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
Lengths of winged turtles. All right. And let's say, and this is all in centimeters. These are very small turtles. So you discover, and these are all adults, so there's a two-centimeter one, there's another two-centimeter one, there's a three-centimeter one, there's another two-centimeter one, there's a five-centimeter... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
These are very small turtles. So you discover, and these are all adults, so there's a two-centimeter one, there's another two-centimeter one, there's a three-centimeter one, there's another two-centimeter one, there's a five-centimeter one, a one-centimeter one, and a six-centimeter one. So we have seven data points. A... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
And from this, and you, I encourage you at any point, if you want, pause this video and see if you wanna calculate, what does the population mean here? We're assuming that this is the population of all the winged turtles. Well, the mean in this situation is going to be equal to, you could add up all of these numbers an... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
And then using these data points and the mean, you can calculate the population standard deviation. And once again, as review, I always encourage you to pause this video and see if you can do it on your own, but I've calculated that ahead of time. The population standard deviation in this situation is approximately, I'... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
So with this information, you should be able to calculate the z-score for each of these data points. Pause this video and see if you can do that. So let me make a new column here. So here I'm gonna put our z-score. And if you just look at the definition, what you're going to do for each of these data points, let's say ... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
So here I'm gonna put our z-score. And if you just look at the definition, what you're going to do for each of these data points, let's say each data point is x, you're going to subtract from that the mean, and then you're going to divide that by the standard deviation. The numerator ID over here is gonna tell you how ... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
So then you'll divide by the population standard deviation. So for example, this first data point right over here, if I wanna calculate its z-score, I will take two. From that, I will subtract three, and then I will divide by 1.69. I will divide by 1.69. And if you got a calculator out, this is going to be negative one... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
I will divide by 1.69. And if you got a calculator out, this is going to be negative one divided by 1.69. And if you use a calculator, you would get this is going to be approximately negative 0.59. And the z-score for this data point is going to be the same. That is also going to be negative 0.59. One way to interpret ... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
And the z-score for this data point is going to be the same. That is also going to be negative 0.59. One way to interpret this is, this is a little bit more than half a standard deviation below the mean. And we could do a similar calculation for data points that are above the mean. Let's say this data point right over ... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
And we could do a similar calculation for data points that are above the mean. Let's say this data point right over here what is its z-score? Pause this video and see if you can figure that out. Well, it's going to be six minus our mean, so minus three. All of that over the standard deviation. All of that over 1.69. An... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
Well, it's going to be six minus our mean, so minus three. All of that over the standard deviation. All of that over 1.69. And this, if you have a calculator, and I calculated it ahead of time, this is going to be approximately 1.77. So more than one, but less than two standard deviations above the mean. I encourage yo... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
And this, if you have a calculator, and I calculated it ahead of time, this is going to be approximately 1.77. So more than one, but less than two standard deviations above the mean. I encourage you to pause this video and now try to figure out the z-scores for these other data points. Now an obvious question that some... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
Now an obvious question that some of you might be asking is why? Why do we care how many standard deviations above or below the mean a data point is? In your future statistical life, z-scores are going to be a really useful way to think about how usual or how unusual a certain data point is. And that's going to be real... | Z-score introduction Modeling data distributions AP Statistics Khan Academy.mp3 |
What we're going to do in this video is start to compare distributions. So for example here we have two distributions that show the various temperatures different cities get during the month of January. This is the distribution for Portland, for example they get eight days between one and four degrees Celsius, they get... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
Now when we make these comparisons, what we're going to focus on is the center of the distributions to compare that, and also the spread. Sometimes people will talk about the variability of the distributions, and so these are the things that we're going to compare. And in making the comparison, we're actually just goin... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
We're not gonna try to pick a measure of central tendency, say the mean or the median, and then calculate precisely what those numbers are for these. We might wanna do those if they're close, but if we can eyeball it, that would be even better. Similar for the spread and variability. In either of these cases, there are... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
In either of these cases, there are multiple measures in our statistical toolkit. Center, mean, median is, mean, median is valuable for the center. For spread variability, the range, the interquartile range, the mean absolute deviation, the standard deviation, these are all measures. But sometimes you can just kind of ... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
But sometimes you can just kind of gauge it by looking. So in this first comparison, which distribution has a higher center, or are they comparable? Well, if you look at the distribution for Portland, the center of this distribution, let's say if we were to just think about the mean, although I think the mean and the m... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
While for Minneapolis, it looks like our center is much closer to maybe negative two or negative three degrees Celsius. So here, even though we don't know precisely what the mean or the median is of each of these distributions, you can say that Portland, Portland distribution has a higher center, has higher center, how... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
Well, if you just superficially thought about range, you see here that there's nothing below one degree Celsius and nothing above 13, so you have about a 13-degree range at most right over here. In fact, what might be contributing to this first column might be a bunch of things at three degrees or even 3.9 degrees, and... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
Let's do another example, and we'll use a different representation for the data here. So we're told at the Olympic Games, many events have several rounds of competition. One of these events is the men's 100-meter backstroke. The upper dot plot shows the times in seconds of the top eight finishers in the final round of ... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
The upper dot plot shows the times in seconds of the top eight finishers in the final round of the 2012 Olympics, so that's in green right over here, the final round. The lower dot plot shows the times of the same eight swimmers but in the semifinal round. So given these distributions, which one has a higher center? We... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
Well, once again, I mean, and here, you can actually, it's a little bit easier to eyeball even what the median might be. The mean, I would probably have to do a little bit more mathematics, but let's say the median, let's say there's one, two, three, four, five, six, seven, eight data points, so the median is gonna sit... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
Well, once again, if you just looked at range, and these are both at the same scale, if you just visually look, the variability here, the range for the final round, is larger than the range for the semifinal round, so you would say that the final round has higher variability, variability, it has a higher range. Eyeball... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
A distribution like this might have a higher range, but lower standard deviation than a distribution like this. Let me just, I'm just drawing a very rough example. A distribution like this has a lower range, but actually might have a higher standard deviation, might have a higher standard deviation than the one above i... | Example Comparing distributions AP Statistics Khan Academy.mp3 |
Researchers used these results to test the null hypothesis is that the proportion is 0.5, the alternative hypothesis is that it's greater than 0.5, where P is the true proportion of adults that support the tax increase. They calculated a test statistic of z is approximately equal to 1.84 and a corresponding P value of ... | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
And we have our four conclusions here. At any point, I encourage you to pause this video and see if you can answer it for yourself. But now we will do it together and just make sure we understand what's going on. Before we even cut to the chase and get to the answer. So what we do is we have this population and we are ... | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
Before we even cut to the chase and get to the answer. So what we do is we have this population and we are going to sample it. So n is equal to 200. From that sample, we calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 support it, which is going to be equal to, let's see, tha... | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
From that sample, we calculate a sample proportion of adults that support the tax increase. We see 113 out of 200 support it, which is going to be equal to, let's see, that is the same thing as 56.5%. So 56.5%. And so the key is is to figure out the P value. What is the probability of getting a result this much above t... | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
And so the key is is to figure out the P value. What is the probability of getting a result this much above the assumed proportion or greater, at least this much above the assumed proportion if we assume that the null hypothesis is true? And if that probability, if that P value is below a preset threshold, if it's belo... | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
If the P value is not lower than this, then we will fail to reject the null hypothesis. Now, to calculate that P value, to calculate that probability, what we figure out is, well, how many, in our sampling distribution, how many standard deviations above the mean of the sampling distribution, and the mean of the sampli... | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
And then we can use this to look at a z-table and say, all right, well, in a normal distribution, what percentage or what is the area under the normal curve that is further than 1.84 standard deviations or at least 1.84 or more standard deviations above the mean? And they give that for us as well. So really, what we ju... | Making conclusions in a test about a proportion AP Statistics Khan Academy.mp3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.