Hypothesis testing with one sample
One-Tailed and Two-Tailed Tests One-Tailed and Two-Tailed Tests
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- In the last video, our null hypothesis was
- the drug had no effect.
- And our alternative hypothesis was that the
- drug just has an effect.
- We didn't say whether the drug would lower the response time
- or raise the response time.
- We just said the drug had an effect, that the mean when you
- have the drug will not be the same thing as
- the population mean.
- And then the null hypothesis says no, your mean with the
- drug's going to be the same thing as the population mean,
- it has no effect.
- In this situation where we're really just testing to see if
- it had an effect, whether an extreme positive effect, or an
- extreme negative effect, would have both been
- considered an effect.
- We did something called a two-tailed test. This is
- called a two-tailed test. Because frankly, a super high
- response time, if you had a response time that was more
- than 3 standard deviations, that would've also made us
- likely to reject the null hypothesis.
- So we were dealing with kind of both tails.
- You could have done a similar type of hypothesis test with
- the same experiment where you only had a one-tailed test.
- And the way we could have done that is we still could have
- had the null hypothesis be that the drug has no effect.
- Or that the mean with the drug-- the mean, and maybe I
- could say the mean with the drug-- is still going to be
- 1.2 seconds, our mean response time.
- Now if we wanted to do a one-tailed test, but for some
- reason we already had maybe a view that this drug would
- lower response times, then our alternative hypothesis-- and
- just so you get familiar with different types of notation,
- some books or teachers will write the alternative
- hypothesis as H1, sometimes they write it as H
- alternative, either one is fine.
- If you want to do one-tailed test, you could say that the
- drug lowers response time.
- Or that the mean with the drug is less than 1.2 seconds.
- Now if you do a one-tailed test like this, what we're
- thinking about is, what we want to look at is, all right,
- we have our sampling distribution.
- Actually, I can just use the drawing that I had up here.
- You had your sampling distribution
- of the sample mean.
- We know what the mean of that was, it's 1.2 seconds, same as
- the population mean.
- We were able to estimate its standard deviation using our
- sample standard deviation, and that was reasonable because it
- had a sample size of greater than 30, so we can still kind
- of deal with a normal distribution for the sampling
- distribution.
- And using that we saw that the result, the sample mean that
- we got, the 1.05 seconds, is 3 standard
- deviations below the mean.
- So if we look at it-- let me just re-draw it with our new
- hypothesis test. So this is the sampling distribution.
- It has a mean right over here at 1.2 seconds.
- And the result we got was 3 standard
- deviations below the mean.
- 1, 2, 3 standard deviations below the mean.
- That was what our 1.05 seconds were.
- So when you set it up like this where you're not just
- saying that the drug has an effect-- in that case, and
- that was the last view, you'd look at both tails.
- But here we're saying we only care is does the drug lower
- our response time?
- And just like we did before, you say OK, let's say the drug
- doesn't lower our response time.
- If the drug doesn't lower our response time, what was the
- probability or what is the probability of getting a
- lowering this extreme or more extreme?
- So here it will only be one of the tails that we could
- consider when we set our alternative hypothesis like
- that, that we think it lowers.
- So if our null hypothesis is true, the probability of
- getting a result more extreme than 1.05 seconds, now we are
- only considering this tail right over here.
- Let me just put it this way.
- More extreme than 1.05 seconds, or let me say, lower.
- Because in the last video we cared about more extreme
- because even a really high result would have said, OK,
- the mean's definitely not 1.2 seconds.
- But in this case we care about means that are lower.
- So now we care about the probability of a result lower
- than 1.05 seconds.
- That's the same thing as sampling-- of getting a sample
- from the sampling distribution that's more than 3 standard
- deviations below the mean.
- And in this case, we're only going to consider the area in
- this one tail.
- So this right here would be a one-tailed test where we only
- care about one direction below the mean.
- If you look at the one-tailed test-- this area over here--
- we saw last time that both of these areas combined are 0.3%.
- But if you're only considering one of these areas, if you're
- only considering this one over here it's going to be half of
- that, because the normal distribution is symmetric.
- So it's going to the 0.13%.
- So this one right here is going to be 0.15%, or if you
- express it as a decimal, this is going to be 0.0015.
- So once again, if you set up your hypotheses like this, you
- would have said, if your null hypothesis is correct, there
- would have only been a 0.15% chance of getting a result
- lower than the result we got.
- So that would be very unlikely, so we will reject
- the null hypothesis and go with the alternative.
- And in this situation your P-value is
- going to be the 0.0015.
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