that are supportable as null hypotheses. So they're,

they're reasonable values of mu. So it isn't that big of a stretch to,

to acknowledge, or to to guess that this will form a confidence interval for mu.

What's interesting is that it forms exactly a

1 minus alpha percent confidence interval for, for mu.

So, if you have a 5% type one error rate for a

set of tests, then you have a 95% confidence interval, which is nice.

and then, the same works in reverse, which

is probably even the more useful direction for us.

If 95% intervals say contains mu not, then we fail to reject mu naught, right?

Which makes sense, right?

The, the value of mu naught was supported as

a potential value when we created the confidence interval.

So it would make sense that we'd elect to, to, to fail to reject H naught.

Then, then, to conclude that mu was different from mu naught.

And, in the next slide, we'll go through the argument.

Okay, let's just briefly go through this argument.

so consider that we do not reject H naught, for a two-sided test

of mu equal to mu naught, versus mu different from mu naught.

If our test statistic, absolute value x bar

minus mu naught divided by the standard error,

s over square root n.

If that's less than the t quantile, the valued it

at 1 minus and n minus 1 degrees of freedom.

And 1 minus alpha over two quantile.

So, you remember, when we reject, if it's bigger than that particular t quantile.

Okay, so we can just, the s over square root of n is positive

so we can just move it over to the right-hand side and we have a

[UNKNOWN]

to the inequality. And this inequality here absolute value of

the x bar minus mu naught less or equal to t, this t quantile times s over square

root of n. And that's exactly equivalent to the

statement below that the mu naught lines is in between x bar minus t times

the square root of standard error. And so and so that is exactly

the same as saying mu naught lies inside the confidence interval.

So this is equivalent to saying, if mu naught lies inside

the confidence interval, then we would have failed to reject H naught.

And then you can obviously reverse this argument to get the other direction.

So that proves the statements we made from the previous slide.

And it

shows that this, sort of inherent duality

between confidence intervals and two-sided hypothesis tests.

This has several uses.

First of all, it, it'll, it tells you if you create say, a 95%

confidence interval, it conveys a little bit

more information than the result of hypothesis test.

Because A, you can do the hypothesis test.