WEBVTT mathematics/statistics/son
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Hi and welcome to www.educator.com.
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We are going to be talking about hypothesis testing today.
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The first thing we need to do is situate ourselves where do hypothesis testing fit in with all of inferential statistics.
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We are going to talk about how to create the hypothesis that we are going to test and that hypothesis is going to be about a population.
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When we say about a population we mean about population parameters.
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There is actually two parts to any hypothesis that we test.
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There is the no hypothesis and the alternative hypothesis.
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We are going to talk about how they fit together.
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We are going to talk about potential errors in hypothesis testing because it is good to know going into it.
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Finally, we are going to end with the steps of hypothesis testing and we are going to do the steps of hypothesis testing,
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When sigma the population standard deviation is given and when it is not given.
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And if you had just refresh yourself with the confidence interval lesson,
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You can probably guess that when sigma is given we are going to be using z distributions or normal distributions.
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When sigma is not given and we have to estimate the population standard deviation from the sample using s then we will use t-distributions.
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In order to use the t distribution we need to figure out the degrees of freedom.
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Let us go back and situate ourselves with all of inferential statistics.
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Basically the idea of inferential statistics is that we use some known populations to figure out the sampling distribution.
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The one that we are using a lot is the SDOM.
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We are going to use the another one later.
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We figure out sampling distributions and now we want to compare a sample from an unknown distribution.
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We want to compare sample from that to the sampling distribution.
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If the sampling distribution says the sample is very likely then we might say maybe the sample,
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this unknown population is very similar to the known population.
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But if the sampling distribution tells us the sample was very unlikely then we could rule out
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the known population as a potential candidate for this unknown population.
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In doing all of this in inferential statistics there are two issues that come up.
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What happens when we do not know what the population looks like at all and
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We want to try to figure out where the population mean or different parameters of the population might be.
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In that case we use confidence intervals and when we use confidence intervals we try to figure out where μ is from x bar.
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Another way of thinking about it is we try to figure out something about the population
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From the sample information because we have that sample information.
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Another technique that we could take is that we could use this idea and say how do we decide when a sample is unlikely?
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How do we decide when to draw x?
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When do we decide this side is weird?
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In order to do that we now have to learn about hypothesis testing.
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The goal of hypothesis testing is to be slightly different from confidence interval yet related.
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It is the flip side of the coin.
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Basically, you are going to try to figure out whether your x bar is unlikely given a hypothetical population.
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In that case, what we are doing is we are setting up a population.
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It is like the population is stable and we are going to compare the sample to it.
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Here is our sample and here is our set standard.
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Here the population is moving but this is the target and this is what we use to get that target.
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Here this is already set and we are comparing this guy to this guy.
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In this way you need both confidence intervals and hypothesis testing to give you the full story.
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You might also hear that hypothesis testing another word or phrase for it will be a test of significance.
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A lot of students misinterpret that to be a test of importance.
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That is the modern way the word significance is used but that is not actually what we are talking about here.
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When we call this at test of significance this is actually using the meaning of significance
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from the early 20th century when this test was actually invented.
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Back then significant adjustment prominence or standing out.
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I like to think of it as being weird like how much does this sample stand out?
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Is that significant?
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Is it prominent and different or is it very, very similar?
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Those are the ways you could think about it.
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I do not want to think of it as a test of importance.
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Now that we know why we need hypothesis testing, how do we hypothesize the population?
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How do we make up a population?
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Do we have to make up all the individual numbers of the population?
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What do we got to do?
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Here is the thing, we could assume things about population parameters and test those assumptions.
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We do not have to stimulate every single member of the population we could just make some assumptions about parameters.
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In order to set up a hypothetical population you set up a parameter.
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For instance, you say μ is equal to something.
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That is how you set up a population then check whether our sample is likely to have come from such a population.
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In doing this we need to figure out how to we hypothesize rigorously so that we could get as much paying for our book from our hypothesis?
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In order to do this we have two parts to a hypothesis and this is going to make our hypothesis better.
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The first part of hypothesis is what we call the null hypothesis and null means 0 or not important.
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The null hypothesis in this case is your hypothetical population.
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We write the null hypothesis like this h sub 0 or h sub knot.
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We might say μ= 0.
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We have created a null hypothesis.
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I just made up to 0 but there are better ways of doing this and we will talk about those later.
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We could also write this in terms of standard deviation or other things but frequently
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you will see the mean being the hypothesis of the population.
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The alternative hypothesis is what do we learn if this is not true?
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If we rule this out then what have we learned?
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In that way these two make up the full hypothesis.
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If we find this then we learn this.
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If we do not find that we learn this other thing.
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What we learn if this is not true is at least that μ does not equal 0.
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This is called the alternative hypothesis and it helps us at least figure out something when we do not figure out that.
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If we do not find this to be true at least we find this to be true.
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If this is not true then we will always find this to be true.
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These two hypotheses together this is more powerful than just having one hypothesis alone.
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We will talk a little bit about why and it goes back to that idea of the test of significance.
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Hypothesis testing or the test of significance is a test of weirdness.
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It tests how weird the x bar is.
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This is the question that it can answer is the x bar weird?
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Is it different from the population?
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But it actually tell is x bar very similar to the population?
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That is not what number gives you but only tells you how weird it is.
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It does not tell you how similar it is.
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These are actually not flip sides of the same coin and because of that our goal here in all
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of hypothesis testing is we find out the most when we reject the null hypotheses.
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That is when we would find out the most.
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This may not seem like we are finding out of luck because we ruled out 0.
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There is an infinite number of μ that we need to test but actually in hypothesis testing
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what you want to do is reject the no rather than accept or fail to reject the null.
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Just because it is set up as a test of weirdness that is the only thing you can find out.
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It is true that it would be nice if we can find out more than that but that is the limitation of this hypothesis testing.
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It is a limitation that is also like the fact of life because even as the limitations this hypothesis testing still a powerful tool.
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But it is good to keep in mind that this one is a limitation.
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A little bit more about these two hypotheses.
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These two hypotheses, the null and the alternative, sometimes you might see the alternative written as h sub 1.
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They must be mutually exclusive.
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This means if one is true the other cannot be true.
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If the other is true, the first cannot be true.
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You cannot have a null hypotheses and alternative hypotheses like μ=1 and μ=2.
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They are not mutually exclusive.
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If one is false, the other one does not have to be true.
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It could be true but it does not have to be.
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Whereas μ does not equal 1, μ = 1.
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Those are mutually exclusive.
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If you rule out one you absolutely know that the other one has to be true.
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Together they must include all possible values of the parameter.
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You can think of the parameters such as μ on a number line and you need to cover the entire number line.
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You can have a null hypothesis like μ > 0.
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You might say μ >0 but then your alternative hypotheses have to be μ < or = 0.
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You color that in and color all of that in too because that is where you will cover the entire space, the parameter space.
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If these are both true, here is what you get.
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One of these two hypotheses must represent the true condition of the population.
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You find out something that is true about the population and then as we said before,
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typically in research your goal is to reject the null and find support for the alternative hypothesis.
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You can actually prove the null hypothesis but you can reject the null hypotheses.
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And the whole reason is because hypothesis testing is a test of significance or test of weirdness.
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This x bar stands out.
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You can only tell me whether it stands out a lot from the population or not.
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They can tell me it is probably similar to the population.
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You cannot tell me that part.
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Let us talk about some errors that we could potentially make in hypothesis testing.
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There are some foibles, you need to watch out for.
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Well first, it helps to imagine that there are two potential realities and we do not know which one of them is true.
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One is that the null hypothesis is true.
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It is actually true.
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We do not know yet, but it is true.
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Other possible reality is that the null hypothesis is false.
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Your sample did not come from the population.
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Those are your two possible realities but only one can be true at any given time.
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You cannot have both the null population being true and false at the same time.
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You got to have one or the other.
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These two boxes, this one and this one have to add up to 100%, but these two boxes , this one and this one have to add up to 1.
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That is because we have a 100% possibility of this being true and 100% possibility of this being true.
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If this is true then this is not true.
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Given that this is reality but we do not know reality, what is the deal?
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How do we put that together with hypothesis testing?
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When we do have hypothesis testing we have 1 of 2 outcomes.
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We could either reject the null successfully, that is what we wanted to do.
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We could either reject the null or we can fail to reject the null.
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We do not call this accept the alternative or accepting the null.
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We call it failing to reject because that is how much we wish we could have rejected the null.
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We failed to reject the null.
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Let us think about these two decisions in conjunction with reality.
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Here is the thing, when we reject the null hypothesis and say this sample did not come from the population.
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If it did not come from that population we would be correct here.
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This would be a correct decision.
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If this is our decision and this is indeed the world we live in, this is a correct decision.
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If we fail to reject the null however but the null is actually true we should not have rejected it
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then this also represents a correct decision.
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Good job not rejecting the null because it is right all along.
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These two are ways that we could be correct.
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That leaves us two ways that we could be incorrect.
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One way is this, we could successfully reject the null but the null is actually true but
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we said that it is false but the null is actually true.
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This is an incorrect decision.
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We call this a false alarm because we are rejecting that now.
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It is false alarm we should have not rejected that null.
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The probability of that false alarm is represented by the term α.
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On the other hand, there is another way that we could be wrong and that way is this.
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We could fail to reject the null.
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We could say we may not be wrong.
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We fail to reject it but the null is wrong.
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This is also an incorrect decision.
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This is not called a false alarm instead it is called a miss.
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This is going to be called the β rate.
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Obviously the α and the β have a probability of less than 1, but greater than 0.
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What we want to do in hypothesis testing is reduce our chance of errors.
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We can also figure out what is our probability of getting different kinds of correct decisions?
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We know that this is one version of the world and that should add up to 100% this probability of failing to reject when we should have kept it around.
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This probability is 1 – α.
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This is what we call a correct failure.
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It sounds odd but it sounds good that you have failed.
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You failed to reject it and you should have failed to reject it.
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It is like you failed to reject a date and you know that date was really good.
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He is a good guy so you should have failed to reject him.
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On the other hand, this is another possible set of what could be right in the world.
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This should add up to 100%, so this should be 1 – β.
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That is our rate of correct decision where we successfully rejected the null and it is indeed false.
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In dating it might be reject somebody who comes up to you and good job you should have rejected them.
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They are a total loser.
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That is what we call a hit.
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It is like in a battleship when you hit it.
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This is the hit rate, miss rate, false alarm rate, and the correct failure rate.
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Let us talk about the steps of hypothesis testing.
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Well there are going to be 5 steps.
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The first step just starts out with setting up your hypothetical population.
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This is the hypothetical population and you need to create both a null hypothesis and an alternative hypothesis then pick a significance level.
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You can think of the word significant as a stand outness like how much it standout.
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How much does it have to standout?
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When it stands out a lot you have a very low false alarm rate.
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If your x bar is out there and then you have a small chance of false alarming.
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You are saying this really does not look like it belongs in the population because it is so out here.
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And that is where your false alarm rate is low.
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You want to set a low one.
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If you want to be more conservative, you want to set an even lower false alarm rate.
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For instance, α = .01 that would be even lower rate of false alarm.
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Then you want to set a decision stage.
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So far, we have not done anything except like setting things up yet and still we are setting things up.
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We set up the decision stage and what you want to do is draw the SDOM, the sampling distribution.
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We have the hypothetical population and we create a sampling distribution so that we can take our sample
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and compare it to that sampling distribution.
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You draw the SDOM and you identify the critical limits.
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Here is my SDOM and you want to identify the extreme regions where you say if your x bar
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is somewhere out here then you want to reject the null.
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You want to say it is very, very unlikely to have come from this null population.
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Then choose a test statistic because the test statistic will tell you how far out from the mean it is in terms of standard error.
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How many jumps out you are?
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This will be called choose a critical test statistics.
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You are saying what are the extreme boundaries such that if x is outside those boundaries we reject it.
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If it is inside the boundaries we do not reject.
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And then we use the sample.
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This is the first time we are doing anything with the sample.
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We use the sample and the SDOM from here to compute the sample test statistic and p value.
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And the p value is going to tell you given that x is out here how much of that curve does it actually cover?
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What is the probability of false alarming there at that particular value?
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And then you compare the sample to this SDOM population and you decide to reject the null or not?
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One word about p value versus α.
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The p value is going to be the probability of belonging to the null population given sample x bar.
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What is the probability that this value belongs in here?
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α is what we call the critical limit.
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This is what we are able to tolerate we just set it.
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α is often decided just by the scientific community.
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In fact α is often set to something like .05 or .01 because that is commonly accepted in scientific communities.
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We call that just being by tradition or convention.
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It is not that we figured out the α level.
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On the other hand we figure out the p value level given our sample x.
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And what we want is for the p value to be lower than the critical limit.
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Let us go through some examples.
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Here is an example of single sample hypothesis testing, also called t tests of 1 mean or single mean t test.
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This is also another term for it.
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Let us talk about this when sigma is available.
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The population standard deviation has been given to us.
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Here it says that the average Www.www.facebook.com.com user has 230 friends, a sigma of 950, a random sample of college students n=39 showed that the sample mean was 393 friends.
00:27:00.500 --> 00:27:04.600
Our college students like the average www.www.facebook.com.com user.
00:27:04.600 --> 00:27:11.100
Let us try to think about this by using hypothesis testing.
00:27:11.100 --> 00:27:23.500
The first thing is perhaps we should set up the best standard population as the average www.www.facebook.com.com user,
00:27:23.500 --> 00:27:28.100
the real population of all Www.www.facebook.com.com users.
00:27:28.100 --> 00:27:35.600
Our null hypothesis might be something like μ= 230.
00:27:35.600 --> 00:27:46.800
That the null hypothesis is that our college students sample is just like everybody else.
00:27:46.800 --> 00:27:58.500
The alternative hypothesis is that our samples are not similar to that population.
00:27:58.500 --> 00:28:03.000
Let us set the significance level.
00:28:03.000 --> 00:28:13.400
Here we could just use α = .05 by convention.
00:28:13.400 --> 00:28:17.900
We could say that is traditional, we will use that too.
00:28:17.900 --> 00:28:21.600
Let us set the decision stage.
00:28:21.600 --> 00:28:31.300
Here we want to start off by drawing the SDOM and I like to label for myself that it is the SDOM
00:28:31.300 --> 00:28:36.700
just so that I do not get confused and mistake it for the population or something like that.
00:28:36.700 --> 00:28:41.000
We want to draw a critical limit.
00:28:41.000 --> 00:28:50.000
If this is the only false alarm that we are willing to tolerate then we might say everything out here we reject.
00:28:50.000 --> 00:28:54.100
Everything out here we reject.
00:28:54.100 --> 00:29:05.100
That would mean that everything in here is 95% and out here these two regions together add up to 5%.
00:29:05.100 --> 00:29:11.400
Because we are going to reject it there is still some probability that this sample belongs to the population.
00:29:11.400 --> 00:29:14.000
But we are going to reject the null.
00:29:14.000 --> 00:29:28.000
We need to split up 5% distributed to both sides so this would make this 2.5% and this would be also 2.5%.
00:29:28.000 --> 00:29:30.800
That is the error that we are going to tolerate.
00:29:30.800 --> 00:29:42.900
I will color n right now my rejection regions so that means if it out here in the extremes I am going to reject my null hypothesis.
00:29:42.900 --> 00:29:51.800
And because we know that this SDOM comes from the population, that is how we are creating this SDOM.
00:29:51.800 --> 00:30:00.700
We know that the μ of SDOM is exactly equal to the μ of the population so that will be 230.
00:30:00.700 --> 00:30:04.800
μ sub x bar = 230.
00:30:04.800 --> 00:30:19.300
We can also figure out the sigma sub x bar and that would be just sigma ÷ √n Which is 950 ÷ √239.
00:30:19.300 --> 00:30:23.200
You could just pull out a calculator to do this.
00:30:23.200 --> 00:30:39.400
I am just going to use the blank Excel file and here is 950 ÷ √239= 61.5.
00:30:39.400 --> 00:30:48.000
That is my standard error of this population.
00:30:48.000 --> 00:30:56.300
And what I want to know is it is nice to have that but if it would also be nice to know what is the z score out here?
00:30:56.300 --> 00:31:01.000
We use z score because we are using sigma.
00:31:01.000 --> 00:31:03.800
What is the z score out here?
00:31:03.800 --> 00:31:15.900
Actually I had just made you memorize it when we previously talked about confidence intervals so we know that is 1.96 and -1.96.
00:31:15.900 --> 00:31:25.200
If you wanted to you could also figure it out by using either the table in the back of your book or Excel
00:31:25.200 --> 00:31:30.600
so we could put in normsin because we have the probability.
00:31:30.600 --> 00:31:42.200
I want the two tailed probability this is actually one tailed.
00:31:42.200 --> 00:31:50.600
The one tailed probability is going to be .025 way down here.
00:31:50.600 --> 00:32:03.900
This little bottom part down here it is covered .025 of this and Excel is telling me that the z score right there is about 1.96.
00:32:03.900 --> 00:32:13.600
Now that we have all of that settled, we could start tinkering with our actual sample.
00:32:13.600 --> 00:32:18.200
Let me draw some space here.
00:32:18.200 --> 00:32:22.000
Let us talk about our sample.
00:32:22.000 --> 00:32:35.400
When we talk about our sample we should figure out how far away is our sample mean?
00:32:35.400 --> 00:32:42.000
We just do not want to know in terms of how far away they are in terms of friends but we want to know
00:32:42.000 --> 00:32:51.300
how far away in terms of the standard deviation because only standard deviation will tell us what proportion of the curve is colored.
00:32:51.300 --> 00:33:02.700
Even if we find out the actual raw distance away 163, we do not know where that is in relation to this curve.
00:33:02.700 --> 00:33:10.300
It would be nice if we could find the z score of 393 then we will know where it is in relation to this curve.
00:33:10.300 --> 00:33:22.100
That would be 393 – 230 so how far is it away from 230, all divided by the standard error 61.5
00:33:22.100 --> 00:33:27.100
because that will give me how many standard errors away we are.
00:33:27.100 --> 00:33:31.200
Let me just calculate that.
00:33:31.200 --> 00:33:52.200
That would be 393 - 230 and I need parentheses because I need it to do the subtraction before the division and that gives me 2.65.
00:33:52.200 --> 00:33:56.300
My z score is 2.65.
00:33:56.300 --> 00:34:08.700
Here this maybe 1 z score away, this is almost 2 z scores away and let us say this is 3 z scores away.
00:34:08.700 --> 00:34:20.800
I know that my 393 is somewhere around here because it is around 2.65.
00:34:20.800 --> 00:34:30.600
This area is very tiny, so I need to find the p value here.
00:34:30.600 --> 00:34:32.100
What is the p value here?
00:34:32.100 --> 00:34:50.700
What is the probability that x bar is greater than or equal to 393?
00:34:50.700 --> 00:35:00.000
That equals the probability that z is greater than or equal to 2.65.
00:35:00.000 --> 00:35:05.900
Not only that but remember we have a two tailed hypothesis.
00:35:05.900 --> 00:35:12.100
We are interested in either being greater than or less than the mean.
00:35:12.100 --> 00:35:16.900
We actually have to find this thing out and multiply it by 2.
00:35:16.900 --> 00:35:28.500
What you can do is look this up in the back of your book and multiply it by 2 or Excel will actually calculate it for you
00:35:28.500 --> 00:35:42.700
like you could put in normsdist and put in the negative side because normsdist gives it to me going from the negative side to positive side.
00:35:42.700 --> 00:35:44.600
I am going to color this part first.
00:35:44.600 --> 00:35:53.500
-2.65 and it should be a very tiny number that will be .004.
00:35:53.500 --> 00:35:59.900
That is a tiny number and then we take that one side and we multiply it by 2 to give us our p value.
00:35:59.900 --> 00:36:19.600
What we are really doing is we are coloring this base, pretend that is inside and also getting -2.65
00:36:19.600 --> 00:36:23.400
and coloring that space and adding those two together.
00:36:23.400 --> 00:36:28.200
That will give us .008.
00:36:28.200 --> 00:36:41.600
What about a single sample hypothesis test when sigma is not available?
00:36:41.600 --> 00:36:48.500
Well this is the exact same problem in fact I have crossed this out so you can no longer use it.
00:36:48.500 --> 00:36:52.200
It is no longer available to you.
00:36:52.200 --> 00:36:59.100
Here what we have to do is estimate sigma and use s instead of sigma.
00:36:59.100 --> 00:37:02.500
Let us go ahead and start off just hypothesis testing.
00:37:02.500 --> 00:37:13.300
Our null hypothesis is μ=230 that are our sample of college students is just like everybody else.
00:37:13.300 --> 00:37:19.400
Our alternative is that they are different from everybody else.
00:37:19.400 --> 00:37:24.500
Different in some way, either have more friends or less friends.
00:37:24.500 --> 00:37:28.600
We also need to pick a significance level.
00:37:28.600 --> 00:37:34.700
How extreme does this x bar have to be?
00:37:34.700 --> 00:37:40.000
We are going to pick α=.05 just by convention we do not figure it out or anything.
00:37:40.000 --> 00:37:44.500
And then we need to set our decision stage.
00:37:44.500 --> 00:37:56.400
Here we want to start off by drawing our SDOM helps to keep this in mind that this is a bunch of means, a bunch of x bars.
00:37:56.400 --> 00:38:03.700
We can just use this information because this is our known population.
00:38:03.700 --> 00:38:08.000
We are going to use that information to figure out our SDOM.
00:38:08.000 --> 00:38:14.500
Here we run into the problem how can we figure out standard error?
00:38:14.500 --> 00:38:22.500
Well, we cannot figure out sigma sub x bar but we can actually figure out s sub x bar.
00:38:22.500 --> 00:38:27.400
That standard error using s instead of sigma.
00:38:27.400 --> 00:38:36.100
That will be s(x) ÷ √n.
00:38:36.100 --> 00:38:46.000
We have s for more sample, the standard deviation of our sample which is 447 ÷ v239.
00:38:46.000 --> 00:39:06.000
And I will just pull out my Excel in order to calculate this.
00:39:06.000 --> 00:39:15.700
447 ÷ v239 and I get 28.9.
00:39:15.700 --> 00:39:35.500
I am actually going to draw in my rejection regions, anything more extreme is going to be rejected.
00:39:35.500 --> 00:39:49.200
Fail to reject in the middle and this rejection region is .025 and this rejection region is .025 because
00:39:49.200 --> 00:39:53.300
I need to split that significance level in 2.
00:39:53.300 --> 00:40:03.700
What we do here is we want to figure out what is our actual t statistic?
00:40:03.700 --> 00:40:08.300
How many standard errors we are when we talk about these borders?
00:40:08.300 --> 00:40:09.800
What is our critical t?
00:40:09.800 --> 00:40:13.000
That would be the t values here.
00:40:13.000 --> 00:40:18.600
This is our raw values in terms of friends but we want to know it in terms of standard error.
00:40:18.600 --> 00:40:26.400
Here are our t values so we cannot just put in 1.96 because that would be for z distributions.
00:40:26.400 --> 00:40:33.800
We need a t distribution and in order to find a t distribution we need degrees of freedom.
00:40:33.800 --> 00:40:42.800
The degrees of freedom is n-1 and that is 238 because 239 – 1.
00:40:42.800 --> 00:40:51.500
You can either look this up in the back of your book or I am going to look this up on Excel.
00:40:51.500 --> 00:41:05.500
Here I am going to use my t inverse and I put in my two tailed probability .05 and my degrees of freedom which is 238.
00:41:05.500 --> 00:41:09.800
And I get 1.97.
00:41:09.800 --> 00:41:24.900
1.97 and -1.97 because t distributions has many problems as they have they are perfectly symmetrical.
00:41:24.900 --> 00:41:27.900
Those are critical t.
00:41:27.900 --> 00:41:31.400
That is the boundary t values.
00:41:31.400 --> 00:41:38.700
Now we have all of that, now we can start thinking about our sample.
00:41:38.700 --> 00:41:45.600
Let us think about our samples t and p value.
00:41:45.600 --> 00:42:02.700
The sample t would be the distance that our sample is away from our mean ÷ standard error because we want how many standard errors away we are.
00:42:02.700 --> 00:42:12.000
393 - 230 ÷ standard error 28.9.
00:42:12.000 --> 00:42:25.800
I will put that into my Excel 393 – 230 ÷ 28.9 = 5.6.
00:42:25.800 --> 00:42:32.200
Let us find the p value there.
00:42:32.200 --> 00:42:40.400
We know that it is far out here our t value so this is about 2, 4, 5.6.
00:42:40.400 --> 00:42:42.400
It is way out here.
00:42:42.400 --> 00:42:45.200
Imagine this going all the way out here.
00:42:45.200 --> 00:42:49.500
That is where x bar landed.
00:42:49.500 --> 00:42:56.800
Already we know that it is pretty far out but let us find the precise p value there.
00:42:56.800 --> 00:43:03.500
In order to find the p value we want to use t dist because that is going to give us the probability.
00:43:03.500 --> 00:43:08.300
We put in the x and that is Excel's word for t.
00:43:08.300 --> 00:43:20.200
When you see x here in t distribution just put in your t value and it only accepts positive t values.
00:43:20.200 --> 00:43:29.000
I will just point to this one, our degrees of freedom which is 238 and how many tails?
00:43:29.000 --> 00:43:31.900
We have a two tailed hypothesis.
00:43:31.900 --> 00:43:43.700
We get 4.8 × 10⁻⁸ so that would be our p value.
00:43:43.700 --> 00:44:17.900
Our probability of getting a t that is greater than or equal to 5.64 or t is less than or equal to -5.64 because it is two tailed equals 4.8 × 10⁻⁸.
00:44:17.900 --> 00:44:29.400
Imagine .07 × 48 and so that is the pretty number.
00:44:29.400 --> 00:44:36.900
This number is so small that they cannot even show you the decimal places.
00:44:36.900 --> 00:44:40.200
It is super close to there but not 0.
00:44:40.200 --> 00:44:45.800
This is our p value, is the p value less than .05?
00:44:45.800 --> 00:44:48.000
Indeed it is.
00:44:48.000 --> 00:44:51.200
What do we do?
00:44:51.200 --> 00:44:54.700
We reject the null hypothesis.
00:44:54.700 --> 00:45:02.000
This is what we do when sigma is not available.
00:45:02.000 --> 00:45:08.700
Just to recap about α versus p value.
00:45:08.700 --> 00:45:20.400
P value is the probability of seeing that sample t or an even more extreme statistic given that the null hypothesis is true.
00:45:20.400 --> 00:45:29.200
And we say extreme because they can be like way bigger or ways smaller either side right.
00:45:29.200 --> 00:45:32.900
α gives you the level of significance.
00:45:32.900 --> 00:45:38.700
That level of extremeness that you have to reach in order to reject your null.
00:45:38.700 --> 00:45:42.200
This is the set standard.
00:45:42.200 --> 00:45:51.000
And this is the thing that you are going to compare to that set standard.
00:45:51.000 --> 00:45:56.700
I want to talk briefly about one versus two-tailed hypotheses.
00:45:56.700 --> 00:46:07.900
When we talk about a one tailed hypothesis, you might have something like μ is going to be greater than 0.
00:46:07.900 --> 00:46:17.100
Or your alternative will be μ is less than 0.
00:46:17.100 --> 00:46:26.200
If that is the case and your set α level is .05 then here is what you would do in your SDOM.
00:46:26.200 --> 00:46:37.700
You will only use one side of it because you are not interested if your x values are way up here.
00:46:37.700 --> 00:46:45.000
You only care if your x value is way smaller than your population.
00:46:45.000 --> 00:46:56.800
In this case, you might set up this as your rejections zone and notice that it only on one side because one tailed and these are end tails.
00:46:56.800 --> 00:47:10.200
That probability will be .05 and this failed to reject side will be .95.
00:47:10.200 --> 00:47:13.400
This is a one tailed hypothesis.
00:47:13.400 --> 00:47:17.800
Frequently we will be dealing with two tailed hypotheses.
00:47:17.800 --> 00:47:24.900
In that case that might be that you do not really care.
00:47:24.900 --> 00:47:33.800
We do not really care if μ is less than, way smaller or way bigger than what we expected.
00:47:33.800 --> 00:47:38.300
We just care if it is extreme in some way, different in some way.
00:47:38.300 --> 00:47:48.400
We do not really care which way and that would be μ = 0 and the alternatives is that μ do not = 0.
00:47:48.400 --> 00:47:59.000
If we had something like α = .05 in a two-tailed hypotheses then we would split up
00:47:59.000 --> 00:48:09.400
that rejection region into the two-tails so that will be .025 and .025.
00:48:09.400 --> 00:48:26.600
We reject , we reject, but inside of these boundaries we fail to reject and this is 90.95%.
00:48:26.600 --> 00:48:35.000
Whatever p value you find we want to compare it to the set α level.
00:48:35.000 --> 00:48:40.200
Let us talk about some examples.
00:48:40.200 --> 00:48:51.400
Your chemistry text book says that if you dissolve table salt and water the freezing point will be lower than it is for pure water 32°f.
00:48:51.400 --> 00:49:00.200
To test this theory, your school does an experiment with 15 teams of students dissolved salt and water and put them in the freezer with the digital thermometer.
00:49:00.200 --> 00:49:04.700
Periodically checking to observe the temperature at which the solution freezes.
00:49:04.700 --> 00:49:08.000
The data is shown in the download below.
00:49:08.000 --> 00:49:11.400
What can you conclude from this data?
00:49:11.400 --> 00:49:23.100
If you look at your download and go to example 1, here are all my freezing temperatures that each of my teams got
00:49:23.100 --> 00:49:26.800
and I think there are only 14 teams here.
00:49:26.800 --> 00:49:29.600
Let us suggest that to be 14.
00:49:29.600 --> 00:49:33.000
What should we do first?
00:49:33.000 --> 00:49:42.300
Just to give you an example of what it is like to do one tailed hypothesis testing, let us have a one tailed test here.
00:49:42.300 --> 00:49:49.900
Because it does say that putting the salt and water the freezing point should be lower
00:49:49.900 --> 00:49:59.100
that automatically gives us a direction that we expect, the freezing point to go in.
00:49:59.100 --> 00:50:04.300
What would our null or default hypothesis be?
00:50:04.300 --> 00:50:10.300
The default hypothesis would be that it is not different from pure water.
00:50:10.300 --> 00:50:11.100
They are the same.
00:50:11.100 --> 00:50:19.100
It might be something like μ=32°f.
00:50:19.100 --> 00:50:27.700
But do we care if our samples are all greater than 32°?
00:50:27.700 --> 00:50:32.000
Maybe the freezing point is higher.
00:50:32.000 --> 00:50:35.500
Do we really care about that?
00:50:35.500 --> 00:50:36.800
No not really.
00:50:36.800 --> 00:50:51.500
Null hypothesis is really that we do not care if it is anything higher than or equal to 32°.
00:50:51.500 --> 00:50:58.600
What we eventually want to know is it lower like weird in this low direction.
00:50:58.600 --> 00:51:11.300
The alternative hypothesis is that it is weird, but in a particular direction that it is too low way lower than 32°.
00:51:11.300 --> 00:51:19.500
Our α is going to be .05, but let us make it clear that it is one tailed.
00:51:19.500 --> 00:51:26.000
Usually they do not say anything but most people assume two tails as the default.
00:51:26.000 --> 00:51:27.700
Let us say one tailed.
00:51:27.700 --> 00:51:36.100
Let us draw this SDOM for the decision stage and here is idea.
00:51:36.100 --> 00:51:53.000
The default is that all the samples come from a population with 32° is the mean of this SDOM but
00:51:53.000 --> 00:52:06.000
we want to know is it weird and a lot lower than that?
00:52:06.000 --> 00:52:08.200
It is consistently lower than that.
00:52:08.200 --> 00:52:23.700
That is our rejection region and that rejection region is going to be .05 because our fail to reject region is going to be .95.
00:52:23.700 --> 00:52:30.400
Now that we have that it would be useful to know what our t statistic here.
00:52:30.400 --> 00:52:36.000
This is raw in terms of degrees Fahrenheit.
00:52:36.000 --> 00:52:38.800
We also want to know the t statistic.
00:52:38.800 --> 00:52:44.000
Here at 0 what is the t statistics here that looks like boundary?
00:52:44.000 --> 00:52:48.300
In order to know that we need to figure out a couple of things.
00:52:48.300 --> 00:52:59.500
I will start with step 3, one of the things I want to know is that t statistics there.
00:52:59.500 --> 00:53:13.600
In order to find that t statistics we need to know degrees of freedom for the sample and that is just account how many axis we have in our sample -1?
00:53:13.600 --> 00:53:16.200
That is 13° of freedom.
00:53:16.200 --> 00:53:19.400
What is the t value there?
00:53:19.400 --> 00:53:32.400
We have the probabilities and we want to know the critical t or boundary t.
00:53:32.400 --> 00:53:40.800
In order to know that we need to use t in here it asks for a two tailed probability.
00:53:40.800 --> 00:53:47.900
We need a one tailed hypothesis so we have to turn that into a two tail probability.
00:53:47.900 --> 00:53:57.600
If this was a two-tailed it would it be .1 and the degrees of freedom is 13.
00:53:57.600 --> 00:54:07.700
It will only give you the positive side, but we could just turn it into -1 because it is perfectly symmetrical.
00:54:07.700 --> 00:54:12.400
This critical t is -1.77.
00:54:12.400 --> 00:54:19.500
Okay, now that we have that, we can start on step 4.
00:54:19.500 --> 00:54:24.100
Step 4 deals with the sample t.
00:54:24.100 --> 00:54:49.200
In order to find the sample t we probably need to find the mean of sample and that is average and we probably also need to know the standard error.
00:54:49.200 --> 00:54:58.700
In order to find standard error what we need is s ÷ √n.
00:54:58.700 --> 00:55:04.100
It is not like for Excel, this is just for me as I need to know s.
00:55:04.100 --> 00:55:08.000
What is my s?
00:55:08.000 --> 00:55:14.500
That would just be stdv in all of these.
00:55:14.500 --> 00:55:26.700
Once I have that then I could calculate standard error s ÷ √n Which is 14.
00:55:26.700 --> 00:55:34.100
We have a standard error, we have a mean, now we can find our sample t
00:55:34.100 --> 00:55:47.200
and that is going to be the mean of the sample - the hypothesized μ 32 ÷ the standard error.
00:55:47.200 --> 00:55:53.900
I get -3.7645.
00:55:53.900 --> 00:56:03.000
We know that this is much more extreme on the negative side than -1.77.
00:56:03.000 --> 00:56:05.800
We also need to find the p value.
00:56:05.800 --> 00:56:09.900
What is the p value there?
00:56:09.900 --> 00:56:15.800
We need to use pdist because we do not know the probability there.
00:56:15.800 --> 00:56:30.000
We put in our t value but remember Excel only accept positive one and I am only going to put so two – is +.
00:56:30.000 --> 00:56:38.200
The degrees of freedom, which is 13 up here and how many tails?
00:56:38.200 --> 00:56:39.300
Just one.
00:56:39.300 --> 00:56:46.800
That is going to be .001 p value.
00:56:46.800 --> 00:56:55.900
Since I have ran out of room I will just write the p value here so p = .001.
00:56:55.900 --> 00:56:59.800
Is that p value smaller than this α?
00:56:59.800 --> 00:57:01.300
Yes, indeed.
00:57:01.300 --> 00:57:04.100
What can we say?
00:57:04.100 --> 00:57:06.200
We can reject the null.
00:57:06.200 --> 00:57:10.200
What can I conclude from this data?
00:57:10.200 --> 00:57:24.700
I can say that this data shows that it is very unlikely to come from the same population as pure water.
00:57:24.700 --> 00:57:29.600
The freezing point of water will have a variation.
00:57:29.600 --> 00:57:41.500
It will have some probability of not being exactly 32 and this deviation on the negative side is much greater than would be expected by chance.
00:57:41.500 --> 00:57:45.100
Let us see.
00:57:45.100 --> 00:57:52.600
Example 2, the heights of women in the United States are approximately normally distributed with a mean of 64.8 in.
00:57:52.600 --> 00:57:59.300
The heights of 11 players on a recent roster of the WNBA team are these in inches.
00:57:59.300 --> 00:58:05.200
Is there sufficient evidence to say that this sample is so much taller than the population that
00:58:05.200 --> 00:58:09.500
this difference cannot reasonably be attributed to chance alone?
00:58:09.500 --> 00:58:13.600
Let us do some hypothesis testing.
00:58:13.600 --> 00:58:20.300
Here our null hypothesis is that our sample is just like regular women.
00:58:20.300 --> 00:58:24.200
The mean is 64.8.
00:58:24.200 --> 00:58:34.000
I am going to use a two tailed alternative here, is that they are not like this population.
00:58:34.000 --> 00:58:42.200
We can probably guess by using common sense that they are on average taller, but we will do a two-tailed test.
00:58:42.200 --> 00:58:45.100
It is actually more conservative.
00:58:45.100 --> 00:58:47.500
It is safer to go with that two tailed test.
00:58:47.500 --> 00:58:55.900
Here we will make α=.05 and it will be two-tailed.
00:58:55.900 --> 00:59:01.800
Let us draw the SDOM here.
00:59:01.800 --> 00:59:24.800
Here we might draw these boundaries and because it is two tailed this is .025 .025 and here it is .95.
00:59:24.800 --> 00:59:28.100
All together it adds up to .1.
00:59:28.100 --> 00:59:35.200
Now that we have this can we figure out the t?
00:59:35.200 --> 00:59:39.100
In order to figure out the t, we need to have the degrees of freedom.
00:59:39.100 --> 00:59:50.100
If you go to the download and go to example 2, I have listed this data here for you and we can actually find the degrees of freedom here.
00:59:50.100 --> 00:59:56.500
Here I put step 3 so that we know where we are.
00:59:56.500 --> 01:00:06.000
In step 3, we need degrees of freedom and that would be count of all of these guys -1.
01:00:06.000 --> 01:00:09.700
We have 11 players 10° of freedom.
01:00:09.700 --> 01:00:13.100
Let us find the critical t.
01:00:13.100 --> 01:00:26.300
The critical t would be t inverse because we know the two tailed probability .05 and the degrees of freedom.
01:00:26.300 --> 01:00:29.600
That gives us the positive critical t.
01:00:29.600 --> 01:00:39.700
That is 2.23 and -2.23 those are our critical boundaries and anything outside of that, we reject the null.
01:00:39.700 --> 01:00:42.700
Let us go to step 4.
01:00:42.700 --> 01:00:46.400
In step 4 we can start dealing with the sample.
01:00:46.400 --> 01:00:56.300
Let us figure out the sample t in order to do that we need the x bar - the μ ÷ standard error.
01:00:56.300 --> 01:01:03.400
We need to know the samples average x bar.
01:01:03.400 --> 01:01:09.100
We also need to know μ and we also need to know standard error.
01:01:09.100 --> 01:01:14.000
Standard errors is going to be s ÷ √n.
01:01:14.000 --> 01:01:18.800
I need to write these things down because it helps me figure out what we need.
01:01:18.800 --> 01:01:20.000
It is like a shopping list.
01:01:20.000 --> 01:01:23.900
Here I need s.
01:01:23.900 --> 01:01:28.400
Now that I have written all these things down I can just calculate them.
01:01:28.400 --> 01:01:48.900
I need the average and μ which I already know from the problem 64.8.
01:01:48.900 --> 01:01:57.700
I need to get my standard error but before I do that I need to get s standard deviation
01:01:57.700 --> 01:02:09.800
and 1 standard deviation I can take that and ÷ the square root of n which is 11.
01:02:09.800 --> 01:02:26.600
That is my standard error and once I have all of these ingredients, I can assemble my t which is x bar – μ ÷ standard error.
01:02:26.600 --> 01:02:35.200
I get 7.97 and that is way higher than 2.2.
01:02:35.200 --> 01:02:43.200
I am pretty sure I can step 5, reject the null.
01:02:43.200 --> 01:02:56.400
If I go back to my problem, then let me see is there is sufficient evidence to say that this sample is so much taller than the population,
01:02:56.400 --> 01:03:00.200
that this difference cannot be reasonably attributed to chance alone.
01:03:00.200 --> 01:03:13.300
I should say yes because when you are way out here, your probability that you belong to this chance distribution is small
01:03:13.300 --> 01:03:22.600
that it is reasonable for us to say that the sample came from a different population.
01:03:22.600 --> 01:03:29.900
Final example, select the best way to complete the sentence.
01:03:29.900 --> 01:03:44.100
The probability that the null hypothesis is true, that is a false alarm rate.
01:03:44.100 --> 01:03:55.200
It is when the null hypothesis is true, but also it is not just that.
01:03:55.200 --> 01:04:09.800
It is not just the possibility that the null hypothesis is true it is that given that you have a particular sample it seems to leave some information.
01:04:09.800 --> 01:04:15.700
It is not quite complete, but it is not entirely false.
01:04:15.700 --> 01:04:19.000
It is just that it does not have the whole truth.
01:04:19.000 --> 01:04:21.100
It does not have the condition.
01:04:21.100 --> 01:04:30.000
Given that you have this particular sample value, the probability that the null hypothesis is false, that is not true.
01:04:30.000 --> 01:04:33.200
Even if you just remember this.
01:04:33.200 --> 01:04:36.900
Remember this column was null is true.
01:04:36.900 --> 01:04:44.900
α is the set one but the p ones are the ones in there.
01:04:44.900 --> 01:04:49.200
That is just not true.
01:04:49.200 --> 01:04:55.100
The probability that an alternative hypothesis is true.
01:04:55.100 --> 01:04:58.300
Actually, we have not talked about that at all.
01:04:58.300 --> 01:05:05.400
We only talked about having a very low possibility that the null hypothesis is true,
01:05:05.400 --> 01:05:09.800
but we have not talked about increasing the probability that the alternative hypothesis is true.
01:05:09.800 --> 01:05:15.500
Beside why would you reject the null when you have a really small t value?
01:05:15.500 --> 01:05:21.600
A small possibility that the alternative hypothesis is true that does not make sense.
01:05:21.600 --> 01:05:34.600
What about the probability of seeing a sample t as extreme as the one given that the null hypothesis is true.
01:05:34.600 --> 01:05:38.000
This is our entire story I can process it now.
01:05:38.000 --> 01:05:55.900
It is not just that the null hypothesis is true, but it also that when you have a certain sample, that also has to be part of the definition of p value.
01:05:55.900 --> 01:06:07.000
The idea is if we have this t value and it is pretty extreme and the null hypothesis is true.
01:06:07.000 --> 01:06:08.600
That is given.
01:06:08.600 --> 01:06:19.600
Given that the null hypothesis is true, what is the possibility of seeing such extreme t value?
01:06:19.600 --> 01:06:21.600
It is very small.
01:06:21.600 --> 01:06:26.500
We are trying to lower our false alarm rate.
01:06:26.500 --> 01:06:33.000
That is the end of one sample hypothesis testing.