Hypothesis Testing and P-values Hypothesis Testing and P-values
Hypothesis Testing and P-values
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- A neurologist is testing the effect of a drug on response
- time by injecting 100 rats with a unit dose of the drug,
- subjecting each to neurological stimulus and
- recording its response time.
- The neurologist knows that the mean response time for rats
- not injected with the drug is 1.2 seconds.
- The mean of the 100 injected rats response times is 1.05
- seconds with the sample standard
- deviation of 0.5 seconds.
- Do you think that the drug has an affect on response time?
- So to do this we're going to set up two hypotheses.
- We're going to say, one, the first hypothesis is we're
- going to call it the null hypothesis, and that is that
- the drug has no effect on response time.
- And your null hypothesis is always going to be-- you can
- view it as a status quo.
- You assume that whatever your researching has no effect.
- So drug has no effect.
- Or another way to think about it is that the mean of the
- rats taking the drug should be the mean with the drug-- let
- me write it this way-- with the mean is still going to be
- 1.2 seconds even with the drug.
- So that's essentially saying it has no effect, because we
- know that if you don't give the drug the mean response
- time is 1.2 seconds.
- Now, what you want is an alternative hypothesis.
- The hypothesis is no, I think the drug
- actually does do something.
- So the alternative hypothesis, right over here, that the drug
- has an effect.
- Or another way to think about it is that the mean does not
- equal 1.2 seconds when the drug is given.
- So how do we think about this?
- How do we know whether we should accept the alternative
- hypothesis or whether we should just default to the
- null hypothesis because the data isn't convincing?
- And the way we're going to do it in this video, and this is
- really the way it's done in pretty much all of science, is
- you say OK, let's assume that the null hypothesis is true.
- If the null hypothesis was true, what is the probability
- that we would have gotten these results with the sample?
- And if that probability is really, really small, then the
- null hypothesis probably isn't true.
- We could probably reject the null hypothesis and we'll say
- well, we kind of believe in the alternative hypothesis.
- So let's think about that.
- Let's assume that the null hypothesis is true.
- So if we assume the null hypothesis is true, let's try
- to figure out the probability that we would have actually
- gotten this result, that we would have actually gotten a
- sample mean of 1.05 seconds with a standard deviation of
- 0.5 seconds.
- So I want to see if we assumed the null hypothesis is true, I
- want to figure out the probability-- and actually
- what we're going to do is not just figure out the
- probability of this, the probability of getting
- something like this or even more extreme than this.
- So how likely of an event is that?
- To think about that let's just think about the sampling
- distribution if we assume the null hypothesis.
- So the sampling distribution is like this.
- It'll be a normal distribution.
- We have a good number of samples, we
- have 100 samples here.
- So this is the sampling distribution.
- It will have a mean.
- Now if we assume the null hypothesis, that the drug has
- no effect, the mean of our sampling distribution will be
- the same thing as the mean of the population
- distribution, which would be equal to 1.2 seconds.
- Now, what is the standard deviation of our sampling
- The standard deviation of our sampling distribution should
- be equal to the standard deviation of the population
- distribution divided by the square root of our sample
- size, so divided by the square root of 100.
- We do not know what the standard deviation of the
- entire population is.
- So what we're going to do is estimate it with our sample
- standard deviation.
- And it's a reasonable thing to do, especially because we have
- a nice sample size.
- The sample size is greater than 100.
- So this is going to be a pretty good approximator.
- This is going to be a pretty good approximator
- for this over here.
- So we could say that this is going to be approximately
- equal to our sample standard deviation divided by the
- square root of 100, which is going to be equal to our
- sample standard deviation is 0.5, 0.5 seconds, and we want
- to divide that by square root of 100 is 10.
- So 0.5 divided by 10 is 0.05.
- So the standard deviation of our sampling distribution is
- going to be-- and we'll put a little hat over it to show
- that we approximated it with-- we approximated the population
- standard deviation with the sample standard deviation.
- So it is going to be equal to 0.5 divided by 10.
- So 0.05.
- So what is the probability-- so let's think
- about it this way.
- What is the probability of getting 1.05 seconds?
- Or another way to think about it is how many standard
- deviations away from this mean is 1.05 seconds, and what is
- the probability of getting a result at least that many
- standard deviations away from the mean.
- So let's figure out how many standard deviations away from
- the mean that is.
- Now essentially we're just figuring out a Z-score, a
- Z-score for this result right over there.
- So let me pick a nice color-- I haven't used orange yet.
- So our Z-score-- you could even do the Z-statistic.
- It's being derived from these other sample statistics.
- So our Z-statistic, how far are we away from the mean?
- Well the mean is 1.2.
- And we are at 1.05, so I'll put that less just so that
- it'll be a positive distance.
- So that's how far away we are.
- And if we wanted it in terms of standard deviations, we
- want to divide it by our best estimate of the sampling
- distribution's standard deviation, which is this 0.05.
- So this is 0.05, and what is this going to be equal to?
- This result right here, 1.05 seconds.
- 1.2 minus 1.05 is 0.15.
- So this is 0.15 in the numerator divided by 0.05 in
- the denominator, and so this is going to be 3.
- So this result right here is 3 standard deviations
- away from the mean.
- So let me draw this.
- This is the mean.
- If I did 1 standard deviation, 2 standard deviations, 3
- standard deviations-- that's in the positive direction.
- Actually let me draw it a little bit
- different than that.
- This wasn't a nicely drawn bell curve, but I'll do 1
- standard deviation, 2 standard deviation, and then 3 standard
- deviations in the positive direction.
- And then we have 1 standard deviation, 2 standard
- deviations, and 3 standard deviations in
- the negative direction.
- So this result right here, 1.05 seconds that we got for
- our 100 rat sample is right over here.
- 3 standard deviations below the mean.
- Now what is the probability of getting a result
- this extreme by chance?
- And when I talk about this extreme, it could be either a
- result less than this or a result of that extreme in the
- positive direction.
- More than 3 standard deviations.
- So this is essentially, if we think about the probability of
- getting a result more extreme than this result right over
- here, we're thinking about this area under the bell
- curve, both in the negative direction or in
- the positive direction.
- What is the probability of that?
- Well we go from the empirical rule that 99.7% of the
- probability is within 3 standard deviations.
- So this thing right here-- you can look it up on a Z-table as
- well, but 3 standard deviation is a nice clean number that
- doesn't hurt to remember.
- So we know that this area right here I'm doing and just
- reddish-orange, that area right over is 99.7%.
- So what is left for these two magenta or pink areas?
- Well if these are 99.7% and both of these combined are
- going to be 0.3%.
- So both of these combined are 0.3-- I should write it this
- way or exactly-- are 0.3%.
- Or is we wrote it as a decimal it would be 0.003 of the total
- area under the curve.
- So to answer our question, if we assume that the drug has no
- effect, the probability of getting a sample this extreme
- or actually more extreme than this is only 0.3%
- Less than 1 in 300.
- So if the null hypothesis was true, there's only a 1 in 300
- chance that we would have gotten a result
- this extreme or more.
- So at least from my point of view this results seems to
- favor the alternative hypothesis.
- I'm going to reject the null hypothesis.
- I don't know 100% sure.
- But if the null hypothesis was true there's only 1 in 300
- chance of getting this.
- So I'm going to go with the alternative hypothesis.
- And just to give you a little bit of some of the name or the
- labels you might see in some statistics or in some research
- papers, this value, the probability of getting a
- result more extreme than this given the null hypothesis is
- called a P-value.
- So the P-value here, and that really just stands for
- probability value, the P-value right over here is 0.003.
- So there's a very, very small probability that we could have
- gotten this result if the null hypothesis was true, so we
- will reject it.
- And in general, most people have some type
- of a threshold here.
- If you have a P-value less than 5%, which means less than
- 1 in 20 shot, let's say, you know what, I'm going to reject
- the null hypothesis.
- There's less than a 1 in 20 chance of getting that result.
- Here we got much less than 1 in 20.
- So this is a very strong indicator that the null
- hypothesis is incorrect, and the drug
- definitely has some effect.
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