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So far, we've used two inferential

techniques, hypothesis tests and confidence intervals.

And it makes sense that the results

of these two techniques agree with each other.

And they will, if we're using equivalent levels of significance and confidence.

So in this video, we're going to

discuss the interplay between significance levels used

in hypothesis testing as well as confidence

levels used in construction of confidence intervals.

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Broadly we

can say that a significance level and a

comp confidence level are complements of each other.

Think about the most commonly used significance level, 5%,

and think about the most commonly used confidence level, 95%.

So there, it is not a coincidence that the sum of those two numbers adds up to one.

They are indeed complements of each other.

However, whether this compliment rule works or not

depends on whether we're doing a one-sided hypothesis

test or two-sided hypothesis test.

So let's look over here at a two-sided

hypothesis test where we have an alpha of 0.05.

If you have an alpha of 0.05, that means that at each tail you can afford to have

about 0.025 of a tail area, so that the total of those tail areas add up to 5%.

And usually, when we're thinking about confidence intervals,

we're always thinking about the middle whatever percent of the distribution.

So if you're thinking about a 95% confidence level,

we're interested in the middle 95% of the normal curve.

Therefore, a two-sided hypothesis test, with alpha

equals 0.05, where the two tail areas

add up to 0.05, is indeed going to be equivalent to a 95% confidence interval.

If we add up the 95% in the middle with the alpha on the two

tails, we get to the one, which is the the entire area under the curve.

So indeed, for two-sided hypothesis tests, the significance level

and the confidence level are complements of each other.

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What happens when we have a one-sided hypothesis test though?

In this case, we're looking at a one-sided hypothesis test with

alpha equals 0.05.

I've chosen to include the the tail area on the higher end.

We could have done it on the lower end as well.

That doesn't matter.

But the important thing is that it's either on one end or the other end, right?

So, we're looking for where our p value could be anything up to 5% and it would

only be in one tail and we could

still reject to, the null hypothesis under this scenario.

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While the lower tail in this figure is of no

interest to us within the framework of the hypothesis test.

Since confidences always have to be symmetric, right?

And the confidence level is always about

the middle whatever percent of the distribution.

And we cant have a confidence interval that only goes a certain amount

of distance in one direction but more in the other direction, we are actually

going to need to think about the 5% in the lower tail, even though we

are not going to use it for any

sort of decision making within the hypothesis test.

So if you have a one-sided hypothesis test with an alpha equal

to 0.05, your equivalent confidence level is actually going to be 90%.

Because we're allowing for the 5% at the

one end that you're interested in, and we're having

to take care of the other 5% at the other end that you're not interested

in, but we need to take into account

so that the confidence interval can be symmetric.

3:34

So in this, case with a one-sided hypothesis

test, we can't anymore really say that the significance

level that we're using and the confidence level

that we're using are complements of each other directly.

We can still use that idea though to figure out how to get from

one to the other, and really, once again, the key is to

always draw your curve and once you draw your curve and mark

what you're interested in, that's going to kind of allow you to

think about do I need to think about the other tail or not?

And if you're doing a one-sided hypothesis test, you will need to think about the

other tail, and then you can easily arrive at the confidence level that you need.

Why is this of importance?

Oftentimes, you may want to use both methods when you're doing

inference, and it doesn't make sense for your methods to not agree with each other.

So, for the most part, if you have

the given significance level and you have accurately determined

the equivalent confidence level, the results from the

two approaches should always equal agree with each other.

4:33

So, to summarize what we've gone through,

a two-sided hypothesis with a threshold of alpha

is equal to a confidence interval with 1 minus alpha.

So, in this case, if your hypothesis test is

two-sided, your confidence level and your significance level are compliments.

A one-sided hypothesis, with a threshold of alpha, is equivalent

to a confidence interval with 1 minus 2 times alpha.

If the null hypothesis is rejected, a confidence interval that agrees

with the result of the hypothesis test should not include the null value.

This, hopefully, makes sense.

Because if you're saying that you're rejecting a null hypothesis, the

null value, then, should simply not be in the confidence interval.

Otherwise, you would be contradicting yourself, saying that

it's a plausible value for the parameter of interest.

Similarly.

If the null hypothesis is failed to be rejected, a confidence

interval that agrees with the result of the

hypothesis test should, indeed, include the null value.

So using what we've learned about the equivalency of the confidence and

significance levels, you can determine which level to do which technique in.

And then, using the two bullet points at the bottom of this slide,

you can determine whether the results of the two techniques agree with each other.

Moral of the story is, if your confidence

interval includes a null value, don't reject it.

If your confidence interval does not include the null

value, then you can go ahead and reject it.