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# Hypothesis testing and p-values

Sal walks through an example about a neurologist testing the effect of a drug to discuss hypothesis testing and p-values. Created by Sal Khan.

Video transcript

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 meaning
of the population distribution, which would
be equal to 1.2 seconds. Now, what is the standard
deviation of our sampling distribution? 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%. 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.