THEY are the same. Read up on what $\displaystyle \alpha$ is under a composite null hypothesis $\displaystyle H_0:\mu\le 530$.
IT is the sup, which will occur when $\displaystyle \mu=530$
It's mentioned in english there.
ON page 519 it's mentioned there and another place too.
It the supremum NOT max. READ the paragraph starting with 'So why did....'
$\displaystyle \alpha = \sup_{\theta\in H_0}P({\rm reject H_0}|\rm{H_0 is true}) $
which occurs at the boundary, since this is an increasing or decreasing function.
So this is same as the simple test.
In that example it is a max since it's a closed interval, extreme value theorem.
BUT if the null hypothesis is $\displaystyle \mu>3$ then you look at $\displaystyle \alpha$ when $\displaystyle \mu =3$.
OK, I have read that page and it clears most of my doubts.
So under simple hypothesis, alpha is defined as on p.491 of Wackerly.
But when under composite hypothesis, we DEFINE alpha by
$\displaystyle
\alpha = \sup_{\theta\in H_0}P({\rm reject H_0}|\rm{H_0 is true})
$, right?? And this is the more general definition of alpha, right?
Let Ho: theta=theta_o
On p.511 of Wackerly, there are statements about that:
(i) Reject Ho if and only if theta_o lies outside the 100(1-alpha)% confidence interval for theta.
(ii) Fail to reject Ho if and only if theta_o lies inside the 100(1-alpha)% confidence interval for theta.
Is this relationship between hypothesis testing and confidence intervals ALWAYS true?
If so, then we can always use confidence intervals to decide whether to reject Ho or not. For every hypothesis testing question, we can answer it simply by computing the confidence intervals. So WHY do we need to develop a whole new theory about hypothesis testing (rejection region, p-value, etc.) if the two theories are completely equivalent?
Thanks for clearing my doubts!
Note: this is also being discussed in other forum
What I was trying to ask is that, if we can draw exactly the SAME conclusions with confidence intervals ONLY, why bother developing a whole new theory in hypothesis testing? We don't need a new rejection region method and p-value method to draw conclusions.
In the example above:
"A random sample of 26 students who are enrolled in School A was taken and their SAT scores were recorded. The sample mean was 548 with a sample standared derivation s=57. The principal of School A claims that the mean SAT scores of students in her school is higher than the mean SAT scores of all the students in City B which is known to be 530. (City B is the city where School A is located) Does the data support the principal's claim? Assume alpha=0.05."
To answer this, we used Ho: μ=530 v.s. Ha: μ>530 before. But actually we can completely forget about Ho and Ha, and simply compute a CONFIDENCE INTERVAL and see if it contains 530 to decide whether or not the data support the principal's claim, right?? So we don't at all need the new techniques and theory of Ch.10.
Are the theories of confidence intervals and hypothesis testing completely equivalent in terms of drawing conclusions?
Thanks for answering!