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Well this is going to be equal to, this is going to be 0.7 squared times 0.3, times 0.3 to the one, two, three, fourth, to the fourth power. Now is this the only way to get two scores in six attempts? No, there's many ways of getting two scores in six attempts. For example, maybe you miss the first one, the first attem... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
For example, maybe you miss the first one, the first attempt, and then you make the second attempt, you score. Then you miss the third attempt. And then you, let's just say you make the fourth attempt. And then you miss the next two. So then you miss and you miss. This is another way to get two scores in six attempts. ... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
And then you miss the next two. So then you miss and you miss. This is another way to get two scores in six attempts. And what's the probability of this happening? Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order. This is going to be 0.3 times 0.7. You have a 30% chance o... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
And what's the probability of this happening? Well as we'll see, it's going to be exactly this, it's just we're multiplying in different order. This is going to be 0.3 times 0.7. You have a 30% chance of missing the first one, a 70% chance of making the second one. And then times 0.3, 30% chance of missing the third, t... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
You have a 30% chance of missing the first one, a 70% chance of making the second one. And then times 0.3, 30% chance of missing the third, times a 70% chance of making the fourth, times a 30% chance for each of the next two misses. If you wanted this exact circumstance, this exact circumstance, this is once again goin... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So for any one of these particular ways to get exactly two scores in six attempts, the probability is going to be this. So the probability of getting exactly two scores in six attempts, well it's going to be any one of these probabilities times the number of ways you can get six scores, times the number of ways you can... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Well, as you can imagine, this is a combinatorics problem. So you could write this as, you could write this, and let me see how I could, you're gonna take six attempts, so you could write this as six choose, or we're trying to, if you're picking from six things, your six attempts, and you're picking two of them, or two... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
And of course, we can write this in kind of the binomial coefficient notation. We could write this as six choose two, six choose two, and we could just apply the formula for combinations. And if this looks completely unfamiliar, I encourage you to look up combinations on Khan Academy, and we go into some detail on the ... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
It actually makes a lot of sense. This is going to be equal to six factorial over two factorial, over two factorial times six minus two factorial. So six minus two, six minus two factorial. I'll do the factorial in green again. Six minus two factorial, and what's this going to be equal to? This is going to be equal to ... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
I'll do the factorial in green again. Six minus two factorial, and what's this going to be equal to? This is going to be equal to six times five times four times three times two, and I'll just throw in the one there, although it doesn't change the value, over two times one, and six minus two is four, so that's going to... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So this right over here is four factorial, so times four times three times two times one. Well, that and that is going to cancel. And then, let's see, six divided by two is three. So this is 15. There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or the... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So this is 15. There's 15 different ways that you could get, that you could pick two things out of six, I guess is one way to say it, or there's 15, did I say 16? There's 15, the sixes and the fives, there's 15 different ways that you could pick two things out of six, or another way of thinking about it is there's 15 d... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Now, the probability for each of those is this right over here. So the probability of exactly two scores in six attempts, well, this is where we deserve a little bit of a drumroll, this is going to be six choose two times 0.7, 0.7 squared. That's, this is two, two, you're gonna make two, you're gonna make two, and then... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Notice, these will necessarily add up to six. So if this right over here was a three, then this right over here would be a three, and then this would be six minus three, or three right over here. And now, what is this value? Well, it's going to be equal to, it's going to be equal to, we have our 15, three times five, s... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
Well, it's going to be equal to, it's going to be equal to, we have our 15, three times five, so we have this business right over here, it's going to be 15 times, times, let's see, in yellow, 0.7 times 0.7 is going to be times 0.49. And let's see, three to the fourth power would be 81. Three to the fourth power would b... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
So, there you go, whatever this number is, and actually, I might as well get a calculator out and calculate it. So this is going to be, this is going to be, let me, so it's 15 times 0.49, times 0.0081, 0.0081, and we get 0.59535. So this is going to be equal to, let me write it down, and actually, maybe I'll, well, I w... | Probability of making 2 shots in 6 attempts Probability and Statistics Khan Academy.mp3 |
We're defining it as the number of independent trials we need to get a success where the probability of success for each trial is lower case p. And we have seen this before when we introduced ourselves to geometric random variables. Now the goal of this video is to think about, well what is the expected value of a geom... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
But in this video we're actually going to prove it to ourselves mathematically. But the expected value of a geometric random variable is gonna be one over the probability of success on any given trial. So now let's prove it to ourselves. So the expected value of any random variable is just going to be the probability w... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
So the expected value of any random variable is just going to be the probability weighted outcomes that you could have. So you could say it is the probability, the probability that our random variable is equal to one times one plus the probability that our random variable is equal to two times two plus, and you get the... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
It will not take on the value zero because you cannot have a success if you have not had a trial yet. But what is this going to be equal to? Well this is going to be equal to, what's the probability that we have a success on our first trial? And actually let me just write it over here. So this is going to be P. What is... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
And actually let me just write it over here. So this is going to be P. What is this going to be? What is the probability that we don't have a success on our first trial but we have one on our second trial? Well this is going to be one minus P, that's the first trial where we don't have a success, times a success on the... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
Well this is going to be one minus P, that's the first trial where we don't have a success, times a success on the second trial. And actually let me do a few more terms here. So let me erase this a little bit, do a few more terms. This is going to be the probability that X equals two, sorry, the probability that X equa... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
This is going to be the probability that X equals two, sorry, the probability that X equals three, times three. And we're going to keep going on and on and on. Well what's this going to be? Well the probability that X equals three is we're going to have to get two unsuccessful trials. And so the probability of two unsu... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
Well the probability that X equals three is we're going to have to get two unsuccessful trials. And so the probability of two unsuccessful trials is one minus P squared. And then one successful trial, just like that. So you get the general idea. So if I wanted to rewrite this, I'm just going to rewrite it to make it a ... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
So you get the general idea. So if I wanted to rewrite this, I'm just going to rewrite it to make it a little bit simpler. So the expected, at least for the purposes of this proof, so the expected value of X is equal to, I'll write this as one P plus two P times one minus P plus three P times one minus P squared. And w... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
And we're going to keep going on and on and on forever like that. So how do we figure out this sum? Well now I'm going to do a little bit of mathematical trickery or gymnastics, but it's all valid. And if any of y'all have seen the proof of taking a infinite geometric series, then we're going to do a very similar techn... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
And if any of y'all have seen the proof of taking a infinite geometric series, then we're going to do a very similar technique. What I'm going to do here is I'm going to think about well what is one minus P times this expected value? So let's do that. So if I say one minus P times the expected value of X, what is that ... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
So if I say one minus P times the expected value of X, what is that going to be equal to? Well I would multiply every one of these terms by one minus P. So one P times one minus P would be one P times one minus P. You would get that right over there. What about two P times one minus P? What would that be equal to? Well... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
What would that be equal to? Well that would be two P times one minus P and now we're going to multiply it by one minus P again so you're going to get one minus P squared. And so I think you see where this is going and we're just going to keep adding and adding and adding from there. So now we're going to do something ... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
So now we're going to do something really fun and interesting at least from a mathematical point of view. If this is equal to that, if the left hand side is equal to the right hand side, let's just subtract this value from both sides. So on the left hand side I would have the expected value of X, that's that, minus thi... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
Minus one minus P times the expected value of X. So I'm just subtracting this from that side but let me subtract this from that side. Well I could subtract this expression from that but this is equivalent so I'm just going to subtract this from that. And so what do I get? Well let's see, I'm going to have one minus P a... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
And so what do I get? Well let's see, I'm going to have one minus P and then if I subtract one P times one minus P from two P times one minus P, well I'm just going to be left with plus one P times one minus P. And then if I subtract this from that, I'm going to be left with one P times one minus P squared. And we're j... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
And so let me simplify this a little bit. If I distribute this negative, this could be plus and then this would be P minus one. And then if we distribute this expected value of X, we get on the left hand side, let me scroll up a little bit, I don't want to squinch it too much. So let's see, we have the expected value o... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
So let's see, we have the expected value of X and then plus P times the expected value of X, P times the expected value of X minus the expected value of X. These cancel out. It's going to be equal to P plus P times one minus P plus P times one minus P squared. And it's going to keep going on and on and on. Well on the ... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
And it's going to keep going on and on and on. Well on the left hand side, all I have is a P times the expected value of X. If I want to solve for the expected value of X, I just divide both sides by P. So I get, and this is kind of neat, through this mathematical gymnastics I now have, I'm just dividing everything by ... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
On the left hand side, I just have the expected value of X. If I divide all of these terms by P, this first term becomes one, the second term becomes one minus P. This third term, if I divide by P, becomes plus one minus P squared, so forth and so on. Now what's cool about this, this is a classic geometric series with ... | Proof of expected value of geometric random variable AP Statistics Khan Academy.mp3 |
He's curious about the difference of opinion between residents in the north and south parts of the city. He obtained separate random samples of voters from each region. Here are the results. So let's see, in the north, 54 out of the 120 said they want the school, 66 said they didn't. In the south, 77 said they wanted t... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
So let's see, in the north, 54 out of the 120 said they want the school, 66 said they didn't. In the south, 77 said they wanted the school, 63 said they didn't. Duncan wants to use these results to construct a 90% confidence interval to estimate the difference in the proportion of residents in these regions who support... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
Assume that all of the conditions for inference have been met. All right, which of the following is a correct 90% confidence interval based on Duncan's samples? So pause this video and see if you can figure that out and you will need a calculator and depending on your calculator, you might need a Z table as well. In a ... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
In a previous video, we introduced the idea of a two-sample Z interval. We talk about the conditions for inference. Lucky for us here, they say the conditions for inference have been met. So we can go straight to calculating the confidence interval. And that confidence interval is going to be the difference between the... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
So we can go straight to calculating the confidence interval. And that confidence interval is going to be the difference between the sample proportions, so P sub S hat, so the sample proportion in the south minus the sample proportion in the north. It's gonna be that difference plus or minus our critical value, Z star,... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
And that is going to be, our estimate is going to be P hat sub S times one minus P hat sub S, all of that over the sample size in the south plus P hat sub N times one minus P hat sub N, all of that over the sample size in the north. Okay, so our sample proportion in the south, I'll later use a calculator to get a decim... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
In the north, this is going to be 54 out of 120, 54 out of 120. What is my critical Z value? Well, here, I'm gonna have to either use a calculator or a Z table. Remember, we have a 90% confidence interval. And so, let me see, I'll draw it right over here. If this is a normal distribution, and you wanna have a 90% confi... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
Remember, we have a 90% confidence interval. And so, let me see, I'll draw it right over here. If this is a normal distribution, and you wanna have a 90% confidence interval, that means you're containing 90% of the distribution, which means each of these tails, well, combined, they would have 10%, but each of them woul... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
And so, I'm gonna look at a Z table that figures out how many standard deviations below the mean do I need to be in order to get 5% right over here. And then, that's going to tell me, well, if I'm that far below or above, that's gonna be my critical Z value. So let me get that Z table out. So I care about 5%, and I'm u... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
So I care about 5%, and I'm using this in a bit of a reverse direction, but let's see, 5%, so this is a little over 5%, getting closer to 5%, even closer to 5%, and now we've gotten right below 5%. So we're gonna be in between this and this. I could just split the difference, and I could just say 1.6, let's just say 1.... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
So this is going to be approximately equal to 1.645, and then, let's see, we know what P hat sub S is, we know what P hat sub N is. In the south, our sample size is 140, and in the north, our sample size is 120. And so now, I just have to type all of this into the calculator, which is gonna get a little hairy, but we w... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
For the sake of time, we'll accelerate this typing into the calculator, but I'm gonna start with calculating the upper bound, and then, we'll calculate the lower bound. And then, I think I've closed all my parentheses, and so I think we're ready to get the upper bound, is going to be equal to 0.218, or approximately 0.... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
And so this one is looking pretty good, 0.202, but let's get the lower bound now. So I got my calculator back, instead of retyping everything, I'm just gonna put a minus here. So I go to second, and just so you see what I'm doing, second, entry, I see the entry back, and then I can just change, I can just change the pa... | Calculating a confidence interval for the difference of proportions AP Statistics Khan Academy.mp3 |
Here is the summary of her results, or here is a summary of her results. And so they give the same data for both of these samples, and once again, they happen to have the same sample size. They don't need to. But over here, instead of giving us a p-value, we've gotten a confidence interval. Una wants to use these resul... | Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3 |
But over here, instead of giving us a p-value, we've gotten a confidence interval. Una wants to use these results to test her null hypothesis that the mean caloric content is the same versus her alternative hypothesis that they are different. Assume that all conditions for inference have been met. Based on the interval... | Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3 |
Based on the interval, what do we know about the corresponding p-value and conclusion at the alpha is equal to 0.01 level of significance? So pause this video and see if you can figure that out. Remember what a 99% confidence interval is. That says that if we construct confidence intervals 100 times, that 99 of those t... | Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3 |
That says that if we construct confidence intervals 100 times, that 99 of those times, we should overlap with the true parameter that we're trying to estimate. In that case, the true parameter is the true difference of these means. Now, when we do a hypothesis test, we always start assuming that the null hypothesis is ... | Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3 |
And so if we assume that the null hypothesis is true, well, another way of writing this null hypothesis, if the two means are equal, that's the same thing as the difference of the means equaling zero. And since we're assuming this, and this is a 99% confidence interval, then 99 out of 100 times that we do this, we shou... | Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3 |
If you take four minus 6.44, you're going to get negative 2.44. So zero is definitely in the interval. And so another way to think about it, we're not in the 1% of the times where we don't overlap. If we were in the 1% of times where we don't overlap with the assumed difference, then we would reject the null hypothesis... | Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3 |
If we were in the 1% of times where we don't overlap with the assumed difference, then we would reject the null hypothesis. Or another way to think about it is our significance level, 0.01 right over here, it's one minus our confidence level right over here. If our 99% confidence interval overlaps, overlaps with mu fro... | Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3 |
And so we could also say that our p-value is greater than our significance level because that is our significance level. And because of that, we fail to reject our null hypothesis. If this did not overlap with our assumed difference in the means, if it did not overlap with zero, then we would be in that one in 100 scen... | Conclusion for a two-sample t test using a confidence interval AP Statistics Khan Academy.mp3 |
If we only have a few experiments, it's very possible that our experimental probability could be different than our theoretical probability or even very different. But as we have many, many more experiments, thousands, millions, billions of experiments, the probability that the experimental and the theoretical probabil... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
This right over here is a simulation created by Macmillan USA. I'll provide the link as an annotation. And what it does is it allows us to simulate many coin flips and figure out the proportion that are heads. So right over here, we can decide if we want our coin to be fair or not. Right now it says that we have a 50% ... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
So right over here, we can decide if we want our coin to be fair or not. Right now it says that we have a 50% probability of getting heads. We can make it unfair by changing this, but I'll stick with the 50% probability. We want to show that on this graph here. We can plot it. And what this says is at a time, how many ... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
We want to show that on this graph here. We can plot it. And what this says is at a time, how many tosses do we want to take? So let's say, let's just start with 10 tosses. So what this is going to do is take 10 simulated flips of coins with each one having a 50% chance of being heads. And then as we flip, we're gonna ... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
So let's say, let's just start with 10 tosses. So what this is going to do is take 10 simulated flips of coins with each one having a 50% chance of being heads. And then as we flip, we're gonna see our total proportion that are heads. So let's just talk through this together. So starting to toss. And so what's going on... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
So let's just talk through this together. So starting to toss. And so what's going on here after 10 flips? So as you see, the first flip actually came out heads. And if you wanted to say, what's your experimental probability after that one flip? You'd say, well, with only one experiment, I got one head. So it looks lik... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
So as you see, the first flip actually came out heads. And if you wanted to say, what's your experimental probability after that one flip? You'd say, well, with only one experiment, I got one head. So it looks like 100% were heads. But in the second flip, it looks like it was a tails because now the proportion that was... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
So it looks like 100% were heads. But in the second flip, it looks like it was a tails because now the proportion that was heads after two flips was 50%. But in the third flip, it looks like it was tails again because now only one out of three, or 33% of the flips have resulted in heads. Now by the fourth flip, we got ... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
Now by the fourth flip, we got a heads again, getting us back to 50th percentile. Now at the fifth flip, it looks like we got another heads. And so now we have three out of five, or 60% being heads. And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausi... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
And so the general takeaway here is when you have one, two, three, four, five, or six experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability. And this even continues all the way until we get to our ninth or 10th tosses. But what happen... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
So now I'm gonna do another, well, let's just do another 200 tosses and see what happens. So I'm just gonna keep tossing here. And you can see, wow, look at this. There was a big run of getting a lot of heads right over here. And then it looks like there's actually a run of getting a bunch of tails right over here, and... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
There was a big run of getting a lot of heads right over here. And then it looks like there's actually a run of getting a bunch of tails right over here, and then a little run of heads, tails, and then another run of heads. And notice, even after 215 tosses, our experimental probability is still reasonably different th... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
So let's do another 200 and see if we can converge these over time. And what we're seeing in real time here should be the law of large numbers. As our number of tosses get larger and larger and larger, the probability that these two are very different goes down and down and down. Yes, you will get moments where you cou... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
Yes, you will get moments where you could even get 10 heads in a row or even 20 heads in a row, but over time, those will be balanced by the times where you're getting disproportionate number of tails. So I'm just gonna keep going. We're now at almost 800 tosses, and you see now we are converging. Now this is, we're go... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
Now this is, we're gonna cross 1,000 tosses soon. And you can see that our proportion here is now 51%. It's getting close now. We're at 50.6%, and I could just keep tossing. This is 1,100. We're gonna approach 1,200 or 1,300 flips right over here. But as you can see, as we get many, many, many more flips, it was actual... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
We're at 50.6%, and I could just keep tossing. This is 1,100. We're gonna approach 1,200 or 1,300 flips right over here. But as you can see, as we get many, many, many more flips, it was actually valuable to see even after 200 flips that there was a difference in the proportion between what we got from the experiment a... | Experimental versus theoretical probability simulation Probability AP Statistics Khan Academy.mp3 |
So just as a bit of review, a lot of what we do in confidence intervals is we are trying to estimate some population parameter. Let's say it's the proportion. Maybe it's the proportion that will vote for a candidate. We can't survey everyone, so we take a sample. And from that sample, maybe we calculate a sample propor... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
We can't survey everyone, so we take a sample. And from that sample, maybe we calculate a sample proportion. And then using this sample proportion, we calculate a confidence interval on either side of that sample proportion. And what we know is that if we do this many, many, many times, every time we do it, we are very... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And what we know is that if we do this many, many, many times, every time we do it, we are very likely to have a different sample proportion. So that'd be sample proportion one, sample proportion two. And every time we do it, we might get, maybe this is sample proportion two. Not only will we get a different, I guess y... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Not only will we get a different, I guess you could say center of our interval, but the margin of error might change because we are using the sample proportion to calculate it. But the first assumption that has to be true and even to make any claims about this confidence interval with confidence is that your sample is ... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
So like with all things with statistics, you really wanna make sure that you're dealing with a random sample and take great care to do that. The second thing that we have to assume, and this is sometimes known as the normal condition, normal condition. Remember, the whole basis behind confidence intervals is we assume ... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
But in order to make that assumption that it's roughly normal, we have this normal condition. And the rule of thumb here is that you would expect per sample more than 10 successes, successes and, successes and failures each, each. So for example, if your sample size was only 10, let's say the true proportion was 50%, o... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Now, because usually when we're doing confidence intervals, we don't even know the true population parameter, what we would actually just do is look at our sample and just count how many successes and how many failures we have. And if we have less than 10 on either one of those, then we are going to have a problem. So ... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And you actually don't even have to say expect because you're going to get a sample and you could just count how many successes and failures you have. If you don't see that, then the normal condition is not met and the statements you make about your confidence interval aren't necessarily going to be as valid. The last ... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Independence condition. And this is the 10% rule. If we are sampling without replacement, and sometimes it's hard to do replacement. If you're serving people who are exiting a store, for example, you can't ask them to go back into the store or it might be very awkward to ask them to go back into the store. And so the i... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
If you're serving people who are exiting a store, for example, you can't ask them to go back into the store or it might be very awkward to ask them to go back into the store. And so the independence condition is that your sample size, so sample, let me just say n, n is less than 10% of the population size. And so let's... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And if you surveyed 1,000 people, well, that was 1% of the population. So you'd feel pretty good that the independence condition is met. And once again, this is valuable when you are sampling without replacement. Now to appreciate how our confidence intervals don't do what we think they're going to do when any of these... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Now to appreciate how our confidence intervals don't do what we think they're going to do when any of these things are broken, and I'll focus on these latter two, the random sample condition. That's super important, frankly, in all of statistics. So let's first look at a situation where our independence condition break... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
So right over here, you can see that we are using our little gumball simulation. And in that gumball simulation, we have a true population proportion, but someone doing these samples might not know that. We're trying to construct confidence interval with a 95% confidence level. And what we've set up here is we aren't r... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And what we've set up here is we aren't replacing. So every member of our sample, we're not looking at it and then putting it back in. We're just gonna take a sample of 200. And I've set up the population so that it's a far larger than 10% of the population. And then when I drew a bunch of samples, so this is a situati... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And I've set up the population so that it's a far larger than 10% of the population. And then when I drew a bunch of samples, so this is a situation where I did almost 1,500 samples here of size 200. What you can see here is the situations where our true population parameter was contained in the confidence interval tha... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And then you see in red the ones where it's not. And as you can see, we are only having a hit, so to speak. The overlap between the confidence interval that we're calculating in the true population parameter is happening about 93% of the time. And this is a pretty large number of samples. This is truly at a 95% confide... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
And this is a pretty large number of samples. This is truly at a 95% confidence level. This should be happening 95% of the time. Similarly, we can look at a situation where our normal condition breaks down. And our normal condition, we can see here that our sample size right here is 15. And actually, if I scroll down a... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
Similarly, we can look at a situation where our normal condition breaks down. And our normal condition, we can see here that our sample size right here is 15. And actually, if I scroll down a little bit, you can see that the simulation even warns me. There are fewer than 10 expected successes. And you can see that when... | Conditions for valid confidence intervals Confidence intervals AP Statistics Khan Academy.mp3 |
We're told that administrators at a school are considering a policy change. They survey a group of students, staff members, and parents about whether or not they agree with the new policy. The following mosaic plot summarizes their results. Which of the following statements can we justify from the mosaic plot? So pause... | Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3 |
Which of the following statements can we justify from the mosaic plot? So pause this video and try to figure this out on your own. Pick which of these statements can be justified, and there could be more than one based on this mosaic plot. All right, now let's work through this together. So before I even look at the ch... | Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3 |
All right, now let's work through this together. So before I even look at the choices, let me see if I can interpret this. So this mosaic plot, what it does above and beyond a segmented bar chart is it gives us the width that shows us how many, for example, students versus staff versus parents were sampled or surveyed.... | Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3 |
And it looks like more than half of the people surveyed were students, and then staff and parents seem similar. In terms of who is agreeing with the policy, so that's that light blue color, it seems like students are not very likely to agree with the policy. It looks like staff is very likely to agree with the policy, ... | Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3 |
So let's see which of these statements are consistent with what we just looked at. Parents were the least likely to agree with the new policy. No, that's not true. The least likely to agree with the new policy, that's students right over here. They were definitely the least likely. The lowest percentage of students are... | Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3 |
The least likely to agree with the new policy, that's students right over here. They were definitely the least likely. The lowest percentage of students are agreeing with the policy, so I don't like that choice. More than half of the total responses came from students. And that does look like the case, because if you v... | Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3 |
More than half of the total responses came from students. And that does look like the case, because if you view this entire width as the total responses, it looks like the student width right over here, that is more than half of it. It looks like it's about 50-something percent or even 60%. So I like this choice right ... | Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3 |
So I like this choice right over here. And then last but not least, there were more total no responses from students than from staff and parents combined. So let's look at the no responses from students. No, that's that darker blue color. The no responses for students is this area right over here. And then the no, so I... | Analyzing mosaic plots Exploring two-variable data AP Statistics Khan Academy.mp3 |
In a previous video, I had worked through this AP Statistics example problem right over here, and then I had it looked over by folks who are familiar with how it is graded, including some people who had graded the AP Statistics exam themselves. And they pointed out a few problems with how I actually wrote things down. ... | Significance test for a proportion free response (part 2 with correction) Khan Academy (2).mp3 |
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