Hypothesis testing with two samples
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Variance of Differences of Random Variables
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Difference of Sample Means Distribution
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Confidence Interval of Difference of Means
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Clarification of Confidence Interval of Difference of Means
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Hypothesis Test for Difference of Means
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Comparing Population Proportions 1
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Comparing Population Proportions 2
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Hypothesis Test Comparing Population Proportions
Variance of Differences of Random Variables Variance of Differences of Random Variables
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- What I want to do in this video is build up some tools
- in our tool kit for dealing with sums and differences of
- random variables.
- So let's say that we have two random variables, x and y, and
- they are completely independent.
- They are independent random variables.
- And I'm just going to go over a little bit
- of a notation here.
- If we wanted to know the expected, or if we talked
- about the expected value of this random variable x, that
- is the same thing as the mean value of this
- random variable x.
- If we talk about the expected the value of y, that is the
- same thing as the mean of y.
- If we talk about the variance of the random variable x, that
- is it the same thing as the expected value of the squared
- distances between our random variable x and its mean.
- And that right there squared.
- So the expected value of these squared differences, and that
- you could also use the notation sigma squared for the
- random variable x.
- This is just a review of things we already know, but I
- just want to reintroduce it because I'll use this to build
- up some of our tools.
- So you do the same thing with this with random variable y.
- The variance of random variable y is the expected
- value of the squared difference between our random
- variable y and the mean of y, or the
- expected value of y, squared.
- And that's the same thing as sigma squared of y.
- There is the variance of y.
- Now you may or may not already know these properties of
- expected values and variances, but I will
- reintroduce them to you.
- And I won't go into some rigorous proof-- actually, I
- think they're fairly easy to digest.
- So one is is that if I have some third random variable,
- let's say I have some third random variable that is
- defined as being the random variable x plus the random
- variable y.
- Let me stay with my colors just so
- everything becomes clear.
- The random variable x plus the random variable y.
- What is the expected value of z going to be?
- The expected the value of z is going to be equal to the
- expected value of x plus y.
- And this is a property of expected values-- I'm not
- going to prove it rigorously right here-- but the expected
- value of x plus the expected value of y, or another way to
- think about this is that the mean of z is going to be the
- mean of x plus the mean of y.
- Or another way to view it is if I wanted to take, let's say
- I have some other random variable.
- I'm running out of letters here.
- Let's say I have the random variable a, and I define
- random variable a to be x minus y.
- So what's its expected value going to be?
- The expected value of a is going to be equal to the
- expected value of x minus y, which is equal to-- you could
- either view it as the expected value of x plus the expected
- value of negative y, or the expected value of x minus the
- expected value of y, which is the same thing as the mean of
- x minus the mean of y.
- So this is what the mean of our random variable a
- would be equal to.
- And all of this is review and I'm going to use this when we
- start talking about the distributions that are sums
- and differences of other distributions.
- Now let's think about what the variance of random variable z
- is and what the variance of random variable a is.
- So the variance of z-- and just to kind of always focus
- back on the intuition, it makes sense.
- If x is completely independent of y and if I have some random
- variable that is the sum of the two, then the expected
- value of that variable, of that new variable, is going to
- be the sum of the expected values of the other two
- because they are unrelated.
- If my expected value here is 5 and my expected value here is
- 7, completely reasonable that my expected value here is 12,
- assuming that they're completely independent.
- Now if we have a situation, so what is the variance of my
- random variable z?
- And once again, I'm not going do a rigorous proof here, this
- is really just a property of variances.
- But I'm going to use this to establish what the variance of
- our random variable a is.
- So if this squared distance on average is some variance, and
- this one is completely independent, it's squared
- distance on average is some distance, then the variance of
- their sum is actually going to be the sum of their variances.
- So this is going to be equal to the variance of random
- variable x plus the variance of random variable y.
- Or another way of thinking about it is that the variance
- of z, which is the same thing as the variance of x plus y,
- is equal to the variance of x plus the variance of random
- variable y.
- Hopefully that make some sense.
- I'm not proving it to you rigorously.
- And you'll see this in a lot of statistics books.
- Now what I want to show you is that the variance of random
- variable a is actually this exact same thing.
- And that's the interesting thing, because you might say,
- hey, why wouldn't it be the difference?
- We had the differences over here.
- So let's experiment with this a little bit.
- The variance-- so I'll just write this-- the variance of
- random variable a is the same thing as the variance of--
- I'll write it like this-- as x minus y, which is equal to--
- you could view it this way-- which is equal to the variance
- of x plus negative y.
- These are equivalent statements.
- So you could view this as being equal to-- just using
- this over here, the sum of these two variances, so it's
- going to be equal to the sum of the variance of x plus the
- variance of negative y.
- Now what I need to show you is that the variance of negative
- y, of the negative of that random variables are going to
- be the same thing as the variance of y.
- So what is the variance of negative y?
- The variance of negative y is the same thing as the variance
- of negative y, which is equal to the expected value of the
- distance between negative y and the expected value of
- negative y squared.
- That's all the variance actually is.
- Now what is the expected value of negative y right over here?
- Actually, even better let me factor out a negative 1.
- So what's in the parentheses right here, this is the exact
- same thing as negative 1 squared times y plus the
- expected value of negative y.
- So that's the same exact same thing in the
- parentheses, squared.
- So everything in magenta is everything in magenta here,
- and it is the expected value of that thing.
- Now what is the expected value of negative y?
- The expected value of negative y-- I'll do it over here-- the
- expected value of the negative of a random variable is just a
- negative of the expected value of that random variable.
- So if you look at this we can re-write this-- I'll give
- myself a little bit more space-- we can re-write this
- as the expected value-- the variance of negative y is the
- expected value-- this is just 1.
- Negative 1 squared is just 1.
- And over here you have y, and instead just write plus the
- expected value of negative y, that's the same thing as minus
- the expected value of y.
- So you have that, and then all of that squared.
- Now notice, this is the exact same thing by definition as
- the variance of y.
- So what we just showed you just now, so this is the
- variance of y.
- So we just showed you is that the variance of the difference
- of two independent random variables is equal to the sum
- of the variances.
- You could definitely believe this, it's equal to the sum of
- the variance of the first one plus the variance of the
- negative of the second one.
- And we just showed that that variance is the same thing as
- the variance of the positive version of that variable,
- which makes sense.
- Your distance from the mean is going to be-- it doesn't
- matter whether you're taking the positive or the negative
- of the variable.
- You just cared about absolute distance.
- So it makes complete sense that that quantity and that
- quantity is going to be the same thing.
- Now the whole reason why I went through this exercise,
- kind of the important takeaways here is that the
- mean of differences right over here-- so I could re-write it
- as the differences of the random variable is the same
- thing as the differences of their means.
- And then the other important takeaway, and I'm going to
- build on this in the next few videos, is that the variance
- of the difference-- if I define a new random variable
- is the difference of two other random variables, the variance
- of that random variable is actually the sum of the
- variances of the two random variables.
- So these are the two important takeaways that we'll use to
- build on in future videos.
- Anyway, hopefully that wasn't too confusing.
- If it was, you can kind of just accept these at face
- value and just assume that these are tools
- that you can use.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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